The fourth stable homotopy group of spheres is trivial, meaning that $\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\in\{2,3,4,5\}$, as we will briefly explain.

Curtis proved that $\pi_n(S^4)\neq 0$ for all $n\geq 4$ \cite{Cur69}. Curtis also proved that $\pi_n(S^2)\neq 0$ for all $n\not\equiv 1\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \cite{MMO75}. The proof that $\pi_n(S^5)\neq 0$ for all $n\geq 5$ was given by Mori \cite{Mor75} and Mahowald \cite{Mah75,Mah82}.

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
    Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

In fact, it appears that every positive integer has at least two Toda primes.

\begin{conj}\label{conj:at least two}
    If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
 Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{conj}\label{conj:at least two}
 If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{figure}[p]
    \includegraphics[width=.75\linewidth]{tn.png}
    \caption{$t(n)$ for $n\leq 100000$}\label{fig:tn}
\end{figure}

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\begin{lem}\label{lem:not 5 or not 3 and 5}
 Let $p\in\{3,5\}$. If $p\nmid n$, then $p\in T(n)$. In particular:
 \begin{itemize}
 \item If $5\nmid n$, then $t(n)\geq 1$.
 \item If $3,5\nmid n$, then $t(n)\geq 2$.
 \end{itemize}
\end{lem}
\begin{proof}
 Let $p\in\{3,5\}$. Then $p-1\mid 4n$, and $\gcd(4n,p)=1$ by assumption. Thus $\gcd(\frac{4n}{p-1},p)=1$, so $p\in T(n)$.
\end{proof}

\begin{lem}\label{lem:t(n)=3}
 Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}
\begin{proof}
 We will induct on $\omega(n)$. Our base cases will consist of $\omega(n)\leq 4$. Note that if $p\nmid n$ for each $p\in\{5,7,13\}$, then $T(n)\supseteq\{5,7,13\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\geq t(a)$ for any prime $p\not\in T(a)$ by \cref{cor:multiply by prime}, so we may assume that every prime factor $p\mid n$ is a Toda prime of some divisor of $n$.
 \begin{itemize}
 \item The case of $\omega(n)=1$ is just the calculation $T(3)=\{5,7,13\}$.
 \item For $\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.
 \item For $\omega(n)=3$, we first compute $t(3\cdot 5\cdot 7)=9$ and $t(3\cdot 5\cdot 13)=t(3\cdot 7\cdot 13)=8$. It remains to compute, for each $p\in\{5,7,13\}$, the Toda primes of $3pq$ for each $q\in T(3p)$. Using the code provided in \cref{sec:code}, we find that $t(3pq)\geq 4$ for all such $p,q$.
 \item For $\omega(n)=4$, we first compute $t(3\cdot 5\cdot 7\cdot 13)=16$. For the remaining computations in this case, we use the code in \cref{sec:code}.
 \begin{itemize}
 \item If $\{p,q\}\subseteq\{5,7,13\}$ and $r\in T(3pq)$, then $t(3pqr)\geq 9$.
 \item If $p\in\{5,7,13\}$ and $\{q,r\}\subseteq T(3p)$, then $t(3pqr)\geq 7$.
 \item If $p\in\{5,7,13\}$, $q\in T(3p)$, and $r\in T(3pq)$, then $t(3pqr)\geq 5$.
 \end{itemize}
 \end{itemize}

\begin{cor}\label{prop:divisible by 3}
 Assume \cref{conj:main}. If $3\mid n$, then $t(n)\geq 3$.
\end{cor}
\begin{proof}
 By \cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \cref{lem:t(n)=3}.
\end{proof}

Firstly, if there exists $q'\in T(3p)-\{5,7,13,q\}$ such that $q'\nmid n$, then $\{5,7,13,q,q'\}-\{p\}\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\{2d+1:d\mid 2n\}$ that are not factors of $n$. Thus if
\begin{equation}\label{eq:heuristic}
\{2d+1\text{ prime}:d\mid 2n\}-(\Omega(n)\cup\{5,7,13,q\})
\end{equation}
is non-empty, then $t(n)\geq 4$. Our heuristic for \cref{conj:main} is that the set $\{2d+1:d\mid 2n\}$ consists of $2^{\omega(n)+1}$ elements, while $\Omega(n)\cup\{5,7,13,q\}$ consists of $\omega(n)+3$ elements.

\begin{prop}\label{prop:t(p)}
Assume $p\geq 7$ is a prime. Let $\vphi$ denote the totient function. If $\vphi(x)=4p$ for some integer $x$, then $T(p)=\{3,5,2p+1\}$ or $\{3,5,4p+1\}$. Otherwise, $T(p)=\{3,5\}$.
\end{prop}
\begin{proof}
 One can directly check that $3,5\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\vphi(x)=p_1^{e_1-1}(p_1-1)\cdots p_m^{e_m-1}(p_m-1)$, where $x=\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\vphi(x)=4p$ if and only if one of the following cases holds:
 \begin{enumerate}[(i)]
 \item $x=2^2\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot 3\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=1$.
 \item $x=2^r\cdot 5^2$, where $r\in\{0,1\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\geq 7$.
 \end{enumerate}
 It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).

\begin{lem}\label{lem:strategy for denoms}
 Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\{2pi+1:i\mid 2a\}$ contains a prime number for each $p\in T(a)$, then \cref{conj:general bernoulli} \eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.
\end{lem}
\begin{proof}
 We know that $p-1\mid 4a$ with $\gcd(\frac{4a}{p-1},p)=1$ for all $p\in T(a)$. Thus $p-1\mid 4am$, and we have $\gcd(\frac{4am}{p-1},p)=1$ if and only if $p\nmid m$. It therefore suffices to show that if $p\mid m$ for some $p\in T(a)$, then $D_{4am}>D_{4a}$.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{lem}\label{lem:t(n)=3}
    Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}

\begin{thm}\label{thm:main}
    Assume \cref{conj:main}. If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.
\end{thm}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\subsection*{Acknowledgements}
We thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.

\label{thm:main}
Assume
\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}
If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.,

Question: For a positive integer m, let T(m) be the set of odd primes p such that p-1 divides 4m and \(\gcd\!\left(p,\frac{4m}{p-1}\right)=1\), and let \(t(m)=|T(m)|\). Assume the following statement is true: whenever m is an odd, square-free multiple of 3, there exists some \(p\in\{5,7,13\}\) with \(p\mid m\), no prime \(r\in\{5,7,13\}\setminus\{p\}\) divides m, and there exists \(q\in T(3p)\setminus\{5,7,13\}\) such that \(q\nmid m\), then \(t(m)\ge 4\). Under this assumption, which statement about a positive integer n holds?

For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

Choices:
B. For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\nmid n\), then \(t(n)\ge 3\).
C. For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\).
D. For every positive integer n, \(t(n)\ge 2\). In addition, if \(5\nmid n\), then \(t(n)\ge 3\); and if \(3\mid n\), then \(t(n)\ge 4\).
E. For every positive integer n, \(t(n)\ge 1\). In addition, if \(3\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

divisibility condition on 3 in the final clause

dropped the strongest conclusion \(3\mid n\Rightarrow t(n)\ge 3\)

replaced the specific hypothesis \(5\nmid n\) by the unrelated \(3\nmid n\)


\begin{proposition}\label{prop:ball-meas-upper-bound}
For any compact set $K\subset \Omega_f$ there exists $r_K>0$ such that
\begin{align*}
    \Ha^n(B_{\Omega_f}(z,r))\le \omega_nd^{n/2}K_IK_O r^n,\quad z\in K,\ r<r_K.
\end{align*}
\end{proposition}

\begin{theorem}\label{thm:higher-integrability}
If $f:\Omega\to \R^n$ is a non-constant quasiregular map of degree $d$, then there exists $p>n$ such that $\minv f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_d(\Omega))$.
\end{theorem}

\begin{theorem}\label{thm:geom-QC}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$, and equip $\Omega_f=\minv f(f(\Omega))$ with the metric from $\mathcal A_d(\R^n)$ and the Hausdorff $n$-measure on $\Omega_f$.
\begin{itemize}
    \item[(1)] $\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\'e inequality;
    \item[(2)] $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal. More precisely,
\begin{align*}
    \frac 1{K_IK_O}\Mod_n\Gamma\le \Mod_n\minv f(\Gamma)\le K_IK_O\Mod_n\Gamma
\end{align*}
fir any path family $\Gamma $ in $f(\Omega)$.
\end{itemize}
\end{theorem}

\begin{theorem}\label{thm:QR-curve}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\minv{f}:f(\Omega)\to \mathcal A_d(\Omega)$ is an $\omega$-quasiregular curve, where $\omega=\operatorname{tr}(\operatorname{vol}_{\R^n})$ is given by \eqref{eq:natural-n-form}. More precisely,  $\minv f\in W_{loc}^{1,n}(f(\Omega),\mathcal A_d(\R^n))$, and
\[
\|\omega\|\circ \minv f\ |D\minv f|^n\le d^{n/2-1}K_I(f)\star \minv f^*\omega\quad \textrm{almost everywhere on }\Omega.
\]
\end{theorem}

\begin{theorem}\label{thm:wug}
For $\Mod_n$-a.e. curve $\gamma$ in $f(\Omega)$, $\minv f\circ\gamma$ is an absolutely continuous curve in $\mathcal{A}_d(\Omega)$, and satisfies
\begin{align}\label{eq:ug-ineq}
H(\gamma_t)|\gamma_t'|\le |(\minv f\circ\gamma)'_t|\le (K_IK_O)^{1/n} H(\gamma_t)|\gamma_t'|
\end{align}
a.e. $t$, where
\begin{align}\label{eq:H}
    H(y):=\big(f_\ast(\|Df\|^{-2})(y)\big)^{1/2},\quad y\in f(\Omega).
\end{align}
\end{theorem}

\begin{lemma}\label{lem:homeo}
The generalized inverse $\minv f:f(X)\to \mathcal A_d(X)$ is a homeomorphism onto its image.
\end{lemma}

\begin{lemma}\label{lem:barycenter}
The barycenter map \eqref{eq:barycenter} is $1/\sqrt d$-Lipschitz.
\end{lemma}

\begin{corollary}\label{cor:QR-curve}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse
\[
g:f(\Omega)\to \R^n,\quad g(y):=\sum_{x\in f\inv(y)}\iota(f,x)x
\]
belongs to $W^{1,p}_{loc}(f(\Omega),\R^n)$.
\end{corollary}

\begin{align}\label{eq:inf-n-PI}
|Dg|_p=\Lip g\quad \mu\textrm{-a.e. on }X
\end{align}

\begin{theorem}\label{thm:sob-pull-back}
Let $n,m,Q\ge 1$ be natural numbers, $U\subset \R^m$ open, and $f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_Q(\R^n))$. If $\omega$ is a $S_Q$-invariant smooth $k$-form on $(\R^n)^Q$, and either $k+1\le p$ of $\ud \omega=0$ and $k\le p$, then $\omega$ has a well-defined pull-back $f^*\omega \in L^{p/k}_{loc}(U,\bigwedge^k\R^m)$ and moreover
\begin{align*}
    \ud(f^*\omega)=f^*(\ud\omega)
\end{align*}
weakly.
\end{theorem}

\subsection{Overview}
A continuous map $f\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\le k$ is a $K$-quasiregular $\omega$-curve, if
\begin{align}\label{eq:QR-curve}
\|\omega\|\circ f\|Df\|^n\le K\star f^*\omega\quad \textrm{ almost everywhere}.
\end{align}
Here, $\omega$ is a (closed) $n$-form on $N$, $\star f^*\omega$ is the Hodge-star dual of $f^*\omega$, and $\|\omega\|$ is the pointwise comass of $\omega$. When $k=n$ and $\omega$ is the volume form on $N$, \eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as
\begin{align}\label{eq:QR}
\|Df\|^n\le K\mathbf{J} f\quad\textrm{ almost everywhere},
\end{align}
where $\mathbf{J} f=\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \cite{vai66}. We call an open discrete map a branched cover.

Quasiregular curves (QR curves) in turn were introduced by Pankka \cite{pankka20}, and they generalize quasiregular maps by allowing the dimension of the target to be higher than that of the domain; for example pseudoholomorhic curves \cite{gro85} are QR curves. Much of the classical theory of QR maps extends to QR curves \cite{pankka20,pan-onn21}, with one notable exception: Reshetnyak's theorem. Namely, QR curves need not be branched covers. Indeed, if $k>n$, a QR curve cannot be open, while the more non-trivial failure of discreteness follows from an example in \cite{IVV02}.

Compared to QR maps, one of the challenges in QR curves is understanding how the choice of $\omega$ reflects the behaviour of $f$, see for example \cite{hei-pan-pry23,iko-pan24} for some rigidity results and connections with calibrations.

\bigskip\noindent In this paper, we identify a fruther connection between QR maps and QR curves: we construct a multi-valued inverse $\minv f$ for a QR map $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$, and show that $\minv f$ is naturally a QR curve (see Theorem \ref{thm:QR-curve}). In particular, QR maps have a local (multi-valued) inverse across the singular branch set (and not only outside it) with suitable Sobolev regularity. Earlier, related constructions include a push-forward operator \cite[p. 263]{HKM06}, \cite[Section 4]{teripekka} and a generalized local inverse \cite[Chapter II.5]{ric93}, \cite[Section 8]{onn-raj09}, both of which are important for obtaining modulus inequalities.

Our approach is based on Almgren's space of unordered tuples \cite{del11} (also known as the symmetric product, cf. \cite[Chapter 4.K]{hat02}), and brings a new perspective to quasiregular theory; together with a pull-back theory by multi-valued maps developed in this paper (see Theorems \ref{thm:sob-pull-back} and \ref{thm:lip-pullback-flat-form}) we obtain e.g. higher integrability properties of the generalized inverse \cite[Chapter II.5]{ric93} (see Corollary \ref{cor:QR-curve} and Theorem \ref{thm:higher-integrability}). We describe the construction and briefly discuss some its advantages over alternative approaches below.

\subsection{Multi-valued inverse}

Given a non-injective (proper) branched cover $f:\Omega\to \mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\Omega)\ni y\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\bar f\circ P_f$, where $P:\Omega\to \widetilde\Omega_f:=\Omega/\sim$ is the projection onto the quotient space given by the equivalence relation $x\sim y$ if $f(x)=f(y)$, and $\bar f:\widetilde \Omega_f\to f(\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\Omega)\to \mathcal{H}(\Omega)$ into the Hausdorff space of non-empty compact subsets of $\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\mathcal{H}(\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \cite[Theorem 2.1]{del11}).

Instead, we give a construction based on Almgren's space and local degree theory. Let $Q\ge 1$ be a natural number. The Almgren space $\mathcal A_Q(\mathbb{R}^n)$ of unordered $Q$-tuples in $\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \cite{alm2000}, and was further explored by De Lellis et. al. \cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as
\begin{align*}
\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q,\quad d_\mathcal A(\bb x,\bb y)=\inf_{\sigma\in S_Q}\Big(\sum_j^d|x_j-y_{\sigma(j)}|^2\Big)^{1/2},
\end{align*}
where $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\mathbb{R}^n)^Q$. (See Section \ref{sec:almgren-space} for more details.)

\bigskip\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\Omega\to \mathbb{R}^n$ of degree $d>0$ from an open set $\Omega\subset \mathbb{R}^n$, the local index $\iota(f,x)$ is defined for all $x\in\Omega$ and satisfies $\displaystyle \sum_{x\in f^{-1}(y)}\iota(f,x)=d$, $y\in f(\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\Omega$ (counted with multiplicities) for each $y\in f(\Omega)$. The multi-valued inverse $\minv f$ of the proper branched cover $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$ is the map
\begin{align}\label{eq:minv}
\minv f:f(\Omega)\to \mathcal A_d(\mathbb{R}^n),\quad \minv f(y):= \sum_{x\in f^{-1}(y)}\iota(f,x)\bb x.
\end{align}
We refer the reader to Section \ref{sec:branched-covers} (see in particular Definition \ref{def:multi-valued-inv}) for the notation and further details used here.

\subsection{Statement of results}

While it is not a manifold, $\mathcal A_Q(\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\mathbb{R}^n$; that is, the $n$-form
\begin{align}\label{eq:natural-n-form}
\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n}):=\sum_j^d P_j^*(\operatorname{vol}_{\mathbb{R}^n})
\end{align}
on $(\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\sigma^*\omega=\omega$ for all $\sigma\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \ref{sec:diff-forms} and \ref{sec:pull-back}), which is of independent interest.

\subsection{Overview}
A continuous map $f\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\le k$ is a $K$-quasiregular $\omega$-curve, if
\begin{align}\label{eq:QR-curve}
\|\omega\|\circ f\|Df\|^n\le K\star f^*\omega\quad \textrm{ almost everywhere}.
\end{align}
Here, $\omega$ is a (closed) $n$-form on $N$, $\star f^*\omega$ is the Hodge-star dual of $f^*\omega$, and $\|\omega\|$ is the pointwise comass of $\omega$. When $k=n$ and $\omega$ is the volume form on $N$, \eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as
\begin{align}\label{eq:QR}
\|Df\|^n\le K\mathbf{J} f\quad\textrm{ almost everywhere},
\end{align}
where $\mathbf{J} f=\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \cite{vai66}. We call an open discrete map a branched cover.

Given a non-injective (proper) branched cover $f:\Omega\to \mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\Omega)\ni y\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\bar f\circ P_f$, where $P:\Omega\to \widetilde\Omega_f:=\Omega/\sim$ is the projection onto the quotient space given by the equivalence relation $x\sim y$ if $f(x)=f(y)$, and $\bar f:\widetilde \Omega_f\to f(\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\Omega)\to \mathcal{H}(\Omega)$ into the Hausdorff space of non-empty compact subsets of $\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\mathcal{H}(\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \cite[Theorem 2.1]{del11}).

Instead, we give a construction based on Almgren's space and local degree theory. Let $Q\ge 1$ be a natural number. The Almgren space $\mathcal A_Q(\mathbb{R}^n)$ of unordered $Q$-tuples in $\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \cite{alm2000}, and was further explored by De Lellis et. al. \cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as
\begin{align*}
\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q,\quad d_\mathcal A(\bb x,\bb y)=\inf_{\sigma\in S_Q}\Big(\sum_j^d|x_j-y_{\sigma(j)}|^2\Big)^{1/2},
\end{align*}
where $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\mathbb{R}^n)^Q$. (See Section \ref{sec:almgren-space} for more details.)

\bigskip\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\Omega\to \mathbb{R}^n$ of degree $d>0$ from an open set $\Omega\subset \mathbb{R}^n$, the local index $\iota(f,x)$ is defined for all $x\in\Omega$ and satisfies $\displaystyle \sum_{x\in f^{-1}(y)}\iota(f,x)=d$, $y\in f(\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\Omega$ (counted with multiplicities) for each $y\in f(\Omega)$. The multi-valued inverse $\minv f$ of the proper branched cover $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$ is the map
\begin{align}\label{eq:minv}
\minv f:f(\Omega)\to \mathcal A_d(\mathbb{R}^n),\quad \minv f(y):= \sum_{x\in f^{-1}(y)}\iota(f,x)\bb x.
\end{align}
We refer the reader to Section \ref{sec:branched-covers} (see in particular Definition \ref{def:multi-valued-inv}) for the notation and further details used here.

\subsection{Statement of results}

While it is not a manifold, $\mathcal A_Q(\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\mathbb{R}^n$; that is, the $n$-form
\begin{align}\label{eq:natural-n-form}
\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n}):=\sum_j^d P_j^*(\operatorname{vol}_{\mathbb{R}^n})
\end{align}
on $(\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\sigma^*\omega=\omega$ for all $\sigma\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \ref{sec:diff-forms} and \ref{sec:pull-back}), which is of independent interest.

\begin{definition}\label{def:multi-valued-inv}
Let $f:X\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\minv f:f(X)\to \mathcal A_d(X)$ of $f$ is defined by
\begin{align*}
\minv f(y)=\sum_{x\in f\inv(y)}\iota(f,x)\bb x
\end{align*}
\end{definition}

\subsection{Statement of results}

In particular, if $f:U\to \mathcal A_Q(\R^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \ref{thm:lip-pullback-flat-form}).

An element $\omega\in \Omega^k(\mathcal A_d(V))$ is a smooth map $\omega:V^d\to \bigwedge^k(\R^n)^d$ with
\begin{align}\label{eq:S_d-invariant}
\omega_x(v_1,\ldots,v_k)=\omega_{\sigma(x)}(\sigma v_1,\ldots,\sigma v_k),\quad x\in V^d,\ v_1,\ldots, v_k\in (\R^n)^d.
\end{align}
In particular, for each $\bb{x}\in \mathcal A_d(V)$, the value $\omega_{\bb{x}}:=\omega_{\sigma(x)}\circ \sigma\in \bigwedge^k(\R^n)^d$
is well-defined and independent of $\sigma\in S_d$. We define the comass $\|\omega\|:\mathcal A_d(V)\to \R$ of a $k$-form $\omega\in \Omega^k(\mathcal A_d(V))$ by
\begin{align*}
\|\omega\|_{\bb x}=\|\omega_{x}\|,\quad \bb x\in \mathcal A_d(V).
\end{align*}
This is well-defined by \eqref{eq:S_d-invariant}.

\begin{lemma}\label{lem:pullback-est}
Suppose $U\subset \R^m$ and $V\subset \R^n$ are open sets, $\omega$ is a smooth $k$-form on $\mathcal A_d(V)$ and $h:U\to \mathcal A_d(V)$ a.e. approximately differentiable. Then $h^*\omega$ is a measurable $k$-form on $U$ and
\begin{align*}
\|h^*\omega\|_x\le \|D_xh\|_x^k\|\omega_{h(x)}\|\quad\textrm{a.e. }x\in U.
\end{align*}
\end{lemma}
\begin{remark}
In particular $h^*\omega\in L^{p/k}_{loc}(U)$ if $h\in W^{1,p}_{loc}(U,\mathcal A_d(V))$.
\end{remark}
\begin{proof}
The approximate differentials $Dh_1,\ldots,Dh_d$ are measurable, cf. ***. For all $x\in U$ where $h$ is approximately differentiable, we have
\begin{align*}
\|h^*\omega\|_x&=\max\{\omega_{h(x)}(Dh(v_1),\ldots,Dh (v_d)):\ |v_1|,\ldots,|v_d|\le 1\}\\
&\le \|D_xh\|^k\|\omega_{h(x)}\|,
\end{align*}
completing the proof.
\end{proof}

\begin{lemma}\label{lem:pull-back-of-decomp}
Suppose $f_i:U\to \mathcal A_{d_i}(V)$ is Lipschitz ($i=0,1$), $d=d_0+d_1$, and $\omega$ is a $S_{d_0}\times S_{d_1}$-invariant form on $V^d$. Then the pull-back $\bb{f_0,f_1}^*\omega$ is a well-defined $k$-form on $U$. Moreover,
\begin{align*}
\bb{f_0,f_1}^*(\omega_0\otimes\omega_1)=f_0^*\omega_0\wedge f_1^*\omega_1
\end{align*}
whenever $\omega_i\in \Omega^{k_i}(\mathcal A_{d_i}(V))$ ($i=0,1$) and $k=k_0+k_1$.
\end{lemma}
\begin{proof}
Let $x\in U$ be such that $f_0,f_1$ are differentiable at $x$, and let $f_i^1(x),\ldots, f_i^{d_i}(x)$ and $D_xf_i^1,\ldots,D_xf_i^{d_i}$ be as in Theorem \ref{thm:multi-valued-rademacher} (i) and (ii). (In particular, note that $f=\bb{f_0,f_1}$ is differentiable at $x$ and $Df=\bb{D_xf_0^1,\ldots,D_xf_0^{d_0},D_xf_1^1,\ldots,D_xf_1^{d_1}}$.) Given $(\sigma_0,\sigma_1)\in S_{d_0}\times S_{d_1}$, denote
\[
D_xf^{(\sigma_0,\sigma_1)}=(D_xf_0^{\sigma_0\inv(1)},\ldots D_xf_0^{\sigma_0\inv(d_0)},D_xf_1^{\sigma_1\inv(1)},\ldots,D_xf_1^{\sigma_1\inv(d_1)})
\]
and similarly
\[
(f_0^{\sigma_0}(x),f_1^{\sigma_1}(x))=(f_0^{\sigma_0\inv(1)}(x),\ldots,f_0^{\sigma_0\inv(d_0)},f_1^{\sigma_1\inv(1)},\ldots,f_1^{\sigma_1\inv(d_1)}).
\]
We define
\begin{align*}
 (f^*\omega)_x(v_1,\ldots,v_k)=\omega_{(f_0^{\sigma_0}(x),f_1^{\sigma_1}(x))}(D_xf^{(\sigma_0,\sigma_1)}v_1,\ldots,D_xf^{(\sigma_0,\sigma_1)}v_k)
\end{align*}
for $v_1,\ldots,v_k\in T_xU=\R^m$. By the $S_{d_0}\times S_{d_1}$-invariance of $\omega$, this expression is independent of $(\sigma_0,\sigma_1)$ and gives a well-defined element in $\bigwedge^k\R^m$, which we denote by $(f^*\omega)_x=\omega_{(f_0(x),f_1(x))}\circ(D_xf_0,D_xf_1)$.

\begin{proposition}\label{prop:flat-form}
Let $d_0,d_1\ge 1$ be natural numbers, $d:=d_0+d_1$, and let $U\subset \R^m$ be an open set. If $f_i:U\to \mathcal A_{d_i}(\R^n)$ are Lipschitz functions such that $\ud(f_i^*\omega)=f_i^*(\ud\omega)$ weakly for all $l$-forms $\omega\in \Omega^l(\mathcal A_{d_i}(\R^n))$, $l\le k$ ($i=0,1$), then $\ud(\bb{f_0,f_1}^*\omega)=\bb{f_0,f_1}^*(\ud\omega)$ weakly for all $k$-forms $\omega\in \Omega^k(\mathcal A_d(\R^n))$.
\end{proposition}
\begin{proof}
Denote $f:=\bb{f_0,f_1}$, and let $\omega\in \Omega^k(\mathcal A_d(\R^n))$. Let
\[
(\omega_j)\subset \bigcup_{r+l=k}\Omega^l((\R^n)^{d_0})\otimes \Omega^r((\R^n)^{d_1})
\]
be a sequence such that $(\omega_j)_x\to \omega_x$ and $(\ud\omega_j)_x\to (\ud\omega)_x$ for all $x\in (\R^n)^d$ and $\sup_j\max\{\|\omega_j\|_\infty,\|\ud\omega_j\|_\infty\}<\infty$. Then
\begin{align*}
\tilde\omega_j&:=P_{S_{d_0}\times S_{d_1}}\omega_j\to P_{S_{d_0}\times S_{d_1}}\omega=\omega,\\
\ud\tilde\omega_j&=P_{S_{d_0}\times S_{d_1}}(\ud\omega_j)\to P_{S_{d_0}\times S_{d_1}}(\ud\omega)=\ud\omega
\end{align*}
and $\sup_j\max\{\|\tilde \omega_j\|_\infty,\|\ud\tilde \omega_j\|_\infty\}<\infty$, cf. Lemma \ref{lem:proj-onto-invariant-forms}. By Lemma \ref{lem:pull-back-of-decomp} each $\tilde\omega_j$ is a sum of tensor products $\omega_0\otimes\omega_1$ where $\omega_i$ is $S_{d_i}$-invariant and $f^*(\omega_0\otimes\omega_1)=(f_0^*\omega_0)\wedge(f^*_1\omega_1)$. Together with the assumption that $f_i^*(\ud\omega_i)=\ud(f^*_i\omega)$ weakly for $S_{d_i}$-invariant forms $\omega$, this yields
\begin{align*}
f^*(\ud(\omega_0\otimes\omega_1))&=f^*(\ud\omega_0\otimes \omega_1+(-1)^{\deg\omega_0}\omega_0\otimes\ud\omega_1)\\
&=f^*_0(\ud\omega_0)\wedge(f^*_1\omega_1)+(-1)^{\deg\omega_0}(f_0^*\omega_0)\wedge (f_1^*(\ud\omega_1))\\
&=\ud(f_0^*\omega_0)\wedge(f_1^*\omega_1)+(-1)^{\deg\omega_0}(f_0^*\omega_0)\wedge\ud(f_1^*\omega_1)\\
&=\ud(f_0^*\omega_0\wedge f_1^*\omega_1)=\ud f^*(\omega_0\otimes\omega_1)
\end{align*}
weakly. Consequently $f^*(\ud\tilde\omega_j)=\ud(f^*\tilde\omega_j)$ weakly. Now the pointwise convergence $f^*\tilde\omega_j\to f^*\omega$, $f^*(\ud\tilde\omega_j)\to f^*(\ud\omega)$ together with the dominated convergence theorem yield
\begin{align*}
\int_U \alpha\wedge f^*(\ud\omega)&=\lim_{j\to\infty}\int_U \alpha\wedge f^*(\ud\tilde\omega_j)=(-1)^{k+1}\lim_{j\to\infty}\int_U\ud\alpha\wedge f^*\tilde\omega_j\\
&=(-1)^{k+1}\int_U\ud\alpha\wedge f^*\omega
\end{align*}
for all $\alpha\in \Omega_c^{m-k-1}(U)$. This proves the claim.
\end{proof}

\begin{proof}
Denote $H(y):=(f_\ast(|Df|^{-2}))^{1/2}$ and observe that
\[
\lim_{h\to 0}\frac{d_\mathcal A(\minv f(y+hv),\minv f(y))}{|h|}=\md_y\minv f(v) \quad\textrm{ a.e. }y\in f(\Omega)
\]
for each $v\in \mathbb S^{n-1}$. By considering the family $\Gamma_v=\{\gamma\subset f(\Omega): \gamma'=v\}$, it follows from a Fubini type argument and Theorem \ref{thm:wug} that
\begin{align}\label{eq:metric-diff-est}
H(y)\le \md_y\minv f(v)\le (K_IK_O)^{1/n}H(y)\quad \textrm{ for all } v\in \mathbb S^{n-1}.
\end{align}
This implies \eqref{eq:jacob-vs-H}. The remaining claim follows from \eqref{eq:metric-diff-est}. Let $\Gamma$ be a path family in $f(\Omega)$ and let $\rho$ be admissible for $\minv f(\Gamma)$. If $\Gamma_0$ is the exceptional family in Theorem \ref{thm:wug} then $\Mod_v\Gamma_0=0$ and, for any $\gamma\in \Gamma\setminus \Gamma_1$ we have
\begin{align*}
1\le \int_{\minv f\circ\gamma}\rho\ud s&=\int_0^1\rho(\minv f(\gamma_t))|(\minv f\circ\gamma)_t'|\ud t\le (K_IK_O)^{1/n}\int_0^1\rho\circ \minv f(\gamma_t)H(\gamma_t)|\gamma_t'|\ud t\\
&=(K_IK_O)^{1/n}\int_\gamma(\rho\circ \minv f)\cdot H\ud s.
\end{align*}
Thus $(K_IK_O)^{1/n}(\rho\circ \minv f)H$ is admissible for $\Gamma\setminus \Gamma_0$ implying
\begin{align*}
\Mod_n\Gamma&=\Mod_n\Gamma\setminus\Gamma_0\le K_IK_O\int_{f(\Omega)}\rho^n\circ \minv f H^n\ud y\\
&\le K_IK_O\int_{f(\Omega)}\rho^n\circ \minv f\J \minv f\ud y=K_IK_O\int_{\Omega_f}\rho^n\ud\Ha^n.
\end{align*}
Taking infimum over $\rho$ yields $\Mod_n\Gamma\le K_IK_O\Mod_n\minv f(\Gamma)$ and completes the proof.
\end{proof}

In particular, if $f:U\to \mathcal A_Q(\mathbb{R}^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \ref{thm:lip-pullback-flat-form}).

Applying the definition of pull-back in the QR context we can show that, if $f:\Omega\to \mathbb{R}^n$ is a quasiregular map, the multi-valued inverse is a quasiregular curve.

\begin{theorem}\label{thm:QR-curve}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\accentset{\leftarrow}{f}:f(\Omega)\to \mathcal A_d(\Omega)$ is an $\omega$-quasiregular curve, where $\omega=\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n})$ is given by \eqref{eq:natural-n-form}. More precisely,  $\minv f\in W_{loc}^{1,n}(f(\Omega),\mathcal A_d(\mathbb{R}^n))$, and
\[
\|\omega\|\circ \minv f\ |D\minv f|^n\le d^{n/2-1}K_I(f)\star \minv f^*\omega\quad \textrm{almost everywhere on }\Omega.
\]
\end{theorem}

Theorems \ref{thm:sob-pull-back} and \ref{thm:QR-curve} together with standard arguments found e.g. in \cite{pan-onn21} yield in particular the higher integrability of the generalized inverse of \cite[Chapter II.5]{ric93}.
\begin{corollary}\label{cor:QR-curve}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse
\[
g:f(\Omega)\to \mathbb{R}^n,\quad g(y):=\sum_{x\in f^{-1}(y)}\iota(f,x)x
\]
belongs to $W^{1,p}_{loc}(f(\Omega),\mathbb{R}^n)$.
\end{corollary}

The multi-valued inverse can also be seen as a homeomorphism $\minv f:f(\Omega)\to \Omega_f$ onto its image $\Omega_f:=\minv f(f(\Omega))\subset \mathcal A_d(\mathbb{R}^n)$ (Lemma \ref{lem:homeo}). Thus, although $\mathcal A_d(\mathbb{R}^n)$ is not a manifold, $\Omega_f$ is a metric $n$-manifold when equipped with the metric from $\mathcal A_d(\mathbb{R}^n)$. In the following theorem we obtain that $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal.

\begin{theorem}\label{thm:geom-QC}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$, and equip $\Omega_f=\minv f(f(\Omega))$ with the metric from $\mathcal A_d(\mathbb{R}^n)$ and the Hausdorff $n$-measure on $\Omega_f$.
\begin{itemize}
    \item[(1)] $\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\'e inequality;
    \item[(2)] $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal. More precisely,
\begin{align*}
    \frac 1{K_IK_O}\Mod_n\Gamma\le \Mod_n\minv f(\Gamma)\le K_IK_O\Mod_n\Gamma
\end{align*}
fir any path family $\Gamma $ in $f(\Omega)$.
\end{itemize}
\end{theorem}

We refer to \eqref{eq:inf-n-PI} in Section \ref{sec:Omega_f-prop} for the definition of infinitesimal Poincar\'e inequality and further commentary. We mention here that the infinitesimal Poincar\'e inequality is not a quantitative condition.

\subsection{Main ideas and further discussion}
Theorem \ref{thm:sob-pull-back} follows by an approximation argument from it Lipschitz counterpart, Theorem \ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. We outline the main idea.

In order to apply induction in the proof of Theorem \ref{thm:lip-pullback-flat-form}, we will need to approximate a given multi-valued Lipschitz map by maps which are locally decomposable. A given map $f:U\to \mathcal A_d(\mathbb{R}^n)$ fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$. The restriction of $f$ to $F$ agrees with the map $d\llbracket b(f)\rrbracket$, where $b:\mathcal A_d(\mathbb{R}^n)\to \mathbb{R}^n$ is the barycenter map
\begin{align}\label{eq:barycenter}
b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d},
\end{align}
which is $\frac 1{\sqrt d}$-Lipschitz, see Lemma \ref{lem:barycenter} below. The equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies that the differentials of the two functions agree almost everywhere. However, since $F$ is closed and not open, we need an approximation argument for the induction step to work. Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$.

The main step in the proof of Theorem \ref{thm:QR-curve} are the Sobolev estimates, which we obtain in Theorem \ref{thm:wug}. Our argument here uses Vitali's covering theorem, which is valid in high generality, and thus Theorem \ref{thm:wug} probably holds e.g. in the setting of \cite{onn-raj09}.

The proof of Theorem \ref{thm:geom-QC} employs a similar induction argument as in Theorem \ref{thm:lip-pullback-flat-form} together with some metric quasiconformal theory. We remark here that a large part of metric quasiconformal theory is based on the Ahlfors regularity of the spaces under consideration. Indeed, in our setting this property would imply the Loewner property of the image set $\Omega_f:=\minv f(f(\Omega)$ by a (deep) result of Semmes \cite{sem96}, which in turn implies the equivalence of the various notions of quasiconformality in the metric setting \cite{hei98}. In this paper, we establish the upper Ahlfors $n$-regularity of $\Omega_f$ (Proposition \ref{prop:ball-meas-upper-bound}), and an infinitesimal Poincar\'e inequality. It would be interesting to analyze the geometry of $\Omega_f$ further, in particular whether $\Omega_f$ admits local bi-Lipschitz parametrizations. By Almgren's bi-Lipschitz embedding $\mathcal A_d(\mathbb{R}^n)\hookrightarrow \mathbb{R}^N$, $\Omega_f$ can be regarded as a subset of Euclidean space, and the existence of bi-Lipschitz parametrizations is linked with the existence of Cartan--Whitney presentations in a suitable Sobolev class  \cite[[Theorem 1.2]{hei11}.

Theorem~\ref{thm:sob-pull-back} "follows by an approximation argument from its Lipschitz counterpart, Theorem~\ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme." The key issue for the induction is that one must "approximate a given multi-valued Lipschitz map by maps which are locally decomposable." A Lipschitz map $f:U\to \mathcal A_d(\mathbb{R}^n)$ "fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$." On $F$, one has that $f$ "agrees with the map $d\llbracket b(f)\rrbracket$, where $b$ is the barycenter map" $b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d}$, and the equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies "the differentials of the two functions agree almost everywhere." Since $F$ is "closed and not open," an approximation is needed: "Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$." This Lipschitz-level result is then used, via approximation, to obtain the Sobolev pull-back statement of Theorem~\ref{thm:sob-pull-back}.

To prove the main theorem, we proceed by an approximation argument from its Lipschitz counterpart. We first prove the following theorem.

\begin{theorem}\label{thm:lip-pullback-flat-form}
Suppose $U\subset\R^m$ is an open set, $f:U\to\mathcal A_d(\R^n)$ is Lipschitz and $\omega\in \Omega^k(\mathcal A_d(\R^n))$. Then $\ud(f^*\omega)=f^*(\ud\omega)$ weakly. In particular, $f^*\omega$ is a flat form on $U$ in the sense of Whitney.
\end{theorem}

Its proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. The key issue for the induction is that one must "approximate a given multi-valued Lipschitz map by maps which are locally decomposable." A Lipschitz map $f:U\to \mathcal A_d(\mathbb{R}^n)$ "fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$." On $F$, one has that $f$ "agrees with the map $d\llbracket b(f)\rrbracket$, where $b$ is the barycenter map" $b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d}$, and the equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies "the differentials of the two functions agree almost everywhere." Since $F$ is "closed and not open," an approximation is needed: "Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$." This Lipschitz-level result is then used, via approximation, in establishing the main theorem.

\label{thm:sob-pull-back}
Let $n,m,Q\ge 1$ be natural numbers, $U\subset \mathbb{R}^m$ open, and $f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_Q(\mathbb{R}^n))$. If $\omega$ is a $S_Q$-invariant smooth $k$-form on $(\mathbb{R}^n)^Q$, and either $k+1\le p$ of $\mathrm {d} \omega=0$ and $k\le p$, then $\omega$ has a well-defined pull-back $f^*\omega \in L^{p/k}_{loc}(U,\bigwedge^k\mathbb{R}^m)$ and moreover
\begin{align*}
    \mathrm {d}(f^*\omega)=f^*(\mathrm {d}\omega)
\end{align*}
weakly.,

Question: Let $n,m,Q\ge 1$ be natural numbers, let $U\subset \mathbb{R}^m$ be open, and let $f\in W^{1,p}_{\mathrm{loc}}(U,\mathcal A_Q(\mathbb{R}^n))$, where $\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q$ is the Almgren space of unordered $Q$-tuples and $S_Q$ acts on $(\mathbb{R}^n)^Q$ by permuting the $Q$ factors. Let $\omega$ be a smooth $k$-form on $(\mathbb{R}^n)^Q$ that is $S_Q$-invariant, meaning $\sigma^*\omega=\omega$ for every permutation $\sigma\in S_Q$. Assume either (i) $k+1\le p$, or (ii) $\mathrm d\omega=0$ and $k\le p$. Under these hypotheses, which conclusion about the pull-back of $\omega$ by $f$ is valid?

The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; moreover its weak exterior derivative satisfies
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$.

Choices:
B.
The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$ whenever $k\le p$; moreover one always has
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$, without any additional assumption such as $k+1\le p$ or $\mathrm d\omega=0$.
C. The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$ under the stated hypotheses.
D.
The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; moreover, if $\mathrm d\omega=0$ and $k\le p$, then
\[
\mathrm d(f^*\omega)=0
\]
weakly on $U$, whereas the identity \(\mathrm d(f^*\omega)=f^*(\mathrm d\omega)\) is guaranteed only in the case $k+1\le p$.
E.
For every smooth $k$-form $\omega$ on $(\mathbb{R}^n)^Q$, the pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; if in addition $\omega$ is $S_Q$-invariant, then one has
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$ under the stated hypotheses.

integrability threshold k+1<=p versus merely k<=p for non-closed forms

differential identity conclusion dropped

closed-form case still yields full chain-rule identity, not just vanishing derivative

S_Q-invariance is required for the pull-back itself to be well defined on Almgren space


\label{thm:trans}
If $x\in \mathcal L_\varepsilon$ for some $\varepsilon>0$, then $x^x$ is transcendental.

A real number $x$ is a \emph{Liouville number} if for every $N\in\mathbb{N}$ there exist integers $p,q$ with $q\ge2$ such that
\[
\bigl|x-\tfrac{p}{q}\bigr|<q^{-N}.
\]
We denote by $\mathcal{L}$ the set of all Liouville numbers. It is classical that $\mathcal{L}$ is dense in $\mathbb{R}$, has Lebesgue measure zero, and is a dense $G_\delta$ subset of $\mathbb{R}$. In particular, $\mathcal{L}$ is uncountable and topologically large while being arithmetically very thin.

In 1962, \cite{erdos} proved that every real number can be expressed both as a sum and, if nonzero, as a product of two Liouville numbers. This also follows from the fact that the set $\mathcal{L}$ of Liouville numbers is a dense $G_{\delta}$ subset of $\mathbb{R}$, although Erd\H{o}s gave an explicit elementary proof. Equivalently, for any nonzero real number $r$, there exist $\sigma,\tau\in\mathcal{L}$ such that $r=f(\sigma,\tau)$ with $f(x,y)=x+y$ or $f(x,y)=xy$.

Later, \cite{burger} extended this phenomenon to a broad family of functions $f(x,y)$. In particular, for $f(x,y)=y^x$, he showed that every positive real number $r\ne1$ can be written as $r=\sigma^{\tau}$ for suitable $\sigma,\tau\in\mathcal{L}$. We refer to~\cite{alniacik1995, Morris2024, Kumar2014} for further topological studies of $\mathcal{L}$ and to~\cite{LG2024} for analogous results concerning $U$-numbers. Consequently, there is no analogue of the Gelfond--Schneider theorem within the class of Liouville numbers. Recently, \cite{ChalebgwaMorris2025} proved that if $S$ is any set of real numbers of full Lebesgue measure, then every positive real number $r$ equals $x^y$, for $x,y\in S$. So, for example, this applies to the case that $S$ is the set of all normal numbers, or the set of real numbers of irrationality exponent 2, or the Mahler set of S-numbers.

Despite this abundance of representations, it remains unknown whether the self-power $x^x$ of a Liouville number $x$ must always be transcendental.

In 2023, \cite{MarquesOliveira2023} introduced a dense $G_\delta$ set (and so with cardinality $\mathfrak{c}$) whose elements have this property. Such numbers are defined as follows:

\begin{definition}
Let $\varepsilon>0$. A real number $x$ is called an \emph{$\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that
\[
\Bigl|x-\frac{p_k}{q_k}\Bigr| < q_k^{-(\log q_k)^{1+\varepsilon}}, \quad \forall k\ge1.
\]
We denote the set of such numbers by $\mathcal L_\varepsilon$.
\end{definition}

\begin{proposition}
Every $\varepsilon$-strong Liouville number is a Liouville number.
\end{proposition}

\begin{proof}
For any $N>0$, take $k$ large enough that $(\log q_k)^{1+\varepsilon} > N$ (e.g., $k>3N^{1/(1+\varepsilon)}+2$). Then
\[
\left|x - \frac{p_k}{q_k}\right| < q_k^{-(\log q_k)^{1+\varepsilon}} < q_k^{-N},
\]
so $x$ is Liouville.
\end{proof}

A real number $x$ is a \emph{Liouville number} if for every $N\in\mathbb{N}$ there exist integers $p,q$ with $q\ge2$ such that
\[
\bigl|x-\tfrac{p}{q}\bigr|<q^{-N}.
\]
We denote by $\mathcal{L}$ the set of all Liouville numbers. It is classical that $\mathcal{L}$ is dense in $\mathbb{R}$, has Lebesgue measure zero, and is a dense $G_\delta$ subset of $\mathbb{R}$. In particular, $\mathcal{L}$ is uncountable and topologically large while being arithmetically very thin.

In 1962, \cite{erdos} proved that every real number can be expressed both as a sum and, if nonzero, as a product of two Liouville numbers. This also follows from the fact that the set $\mathcal{L}$ of Liouville numbers is a dense $G_{\delta}$ subset of $\mathbb{R}$, although Erd\H{o}s gave an explicit elementary proof. Equivalently, for any nonzero real number $r$, there exist $\sigma,\tau\in\mathcal{L}$ such that $r=f(\sigma,\tau)$ with $f(x,y)=x+y$ or $f(x,y)=xy$.

Later, \cite{burger} extended this phenomenon to a broad family of functions $f(x,y)$. In particular, for $f(x,y)=y^x$, he showed that every positive real number $r\ne1$ can be written as $r=\sigma^{\tau}$ for suitable $\sigma,\tau\in\mathcal{L}$. We refer to~\cite{alniacik1995, Morris2024, Kumar2014} for further topological studies of $\mathcal{L}$ and to~\cite{LG2024} for analogous results concerning $U$-numbers. Consequently, there is no analogue of the Gelfond--Schneider theorem within the class of Liouville numbers. Recently, \cite{ChalebgwaMorris2025} proved that if $S$ is any set of real numbers of full Lebesgue measure, then every positive real number $r$ equals $x^y$, for $x,y\in S$. So, for example, this applies to the case that $S$ is the set of all normal numbers, or the set of real numbers of irrationality exponent 2, or the Mahler set of S-numbers.

\begin{definition}
Let $\varepsilon>0$. A real number $x$ is called an \emph{$\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that
\[
\Bigl|x-\frac{p_k}{q_k}\Bigr| < q_k^{-(\log q_k)^{1+\varepsilon}}, \quad \forall k\ge1.
\]
We denote the set of such numbers by $\mathcal L_\varepsilon$.
\end{definition}

\begin{proposition}
Every $\varepsilon$-strong Liouville number is a Liouville number.
\end{proposition}

\begin{proof}
For any $N>0$, take $k$ large enough that $(\log q_k)^{1+\varepsilon} > N$ (e.g., $k>3N^{1/(1+\varepsilon)}+2$). Then
\[
\left|x - \frac{p_k}{q_k}\right| < q_k^{-(\log q_k)^{1+\varepsilon}} < q_k^{-N},
\]
so $x$ is Liouville.
\end{proof}

\begin{definition}
Let $\varepsilon>0$. A real number $x$ is called an \emph{$\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that
\[
\Bigl|x-\frac{p_k}{q_k}\Bigr| < q_k^{-(\log q_k)^{1+\varepsilon}}, \quad \forall k\ge1.
\]
We denote the set of such numbers by $\mathcal L_\varepsilon$.
\end{definition}

\begin{proof}
For any $N>0$, take $k$ large enough that $(\log q_k)^{1+\varepsilon} > N$ (e.g., $k>3N^{1/(1+\varepsilon)}+2$). Then
\[
\left|x - \frac{p_k}{q_k}\right| < q_k^{-(\log q_k)^{1+\varepsilon}} < q_k^{-N},
\]
so $x$ is Liouville.
\end{proof}

\begin{proof}
The proof relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}. We refer the reader to \cite[Proposition 3.2]{MarquesOliveira2023} for details.
\end{proof}

\begin{theorem}
Every spiffy constant belongs to $\mathcal L_\varepsilon$ for all $\varepsilon>0$.
\end{theorem}

\begin{proof}
Let $x=x_a$ be a spiffy constant and define the rational truncations
\begin{equation}
\label{eq:trunc}
r_m = \sum_{n=1}^m \frac{a_n}{3^{e_n}} = \frac{p_m}{q_m}, \qquad q_m = 3^{e_m}.
\end{equation}
Then
\[
|x - r_m| \le q_m^{-3^{e_m-1}}.
\]
Since $\log q_m = e_m \log 3$, we have $(\log q_m)^{1+\varepsilon} = o(3^{e_m-1})$, and therefore, for sufficiently large $m$,
\[
|x - r_m| < q_m^{-(\log q_m)^{1+\varepsilon}}.
\]
Hence, $x \in \mathcal{L}_\varepsilon$ for every $\varepsilon>0$.
\end{proof}

\begin{theorem}\label{thm:strong-Liouville-selfpowers-nonLiouville}
For every $\varepsilon>0$ there exist continuum many $\varepsilon$-strong
Liouville numbers $x$ such that $x^x$ is transcendental and not a Liouville
number.

For any $\xi\in C\setminus\mathcal{B}$, we have $|\xi^\xi-a/b|\geq b^{-\tau}$ for all but finitely many rationals $a/b$, so $\xi^\xi$ is not Liouville. Moreover, since every $\xi\in C$ is $\varepsilon$-strong Liouville, Theorem 1.2 implies $\xi^\xi$ is transcendental. Thus $C\setminus\mathcal{B}$ consists of continuum many $\varepsilon$-strong Liouville numbers $\xi$ for which $\xi^\xi$ is transcendental and not Liouville.
\end{proof}

Having settled that:
\begin{enumerate}
\item for $x$ an $\varepsilon$-strong Liouville number, $x^x$ is a transcendental number;
\item the set of $\varepsilon$-strong Liouville numbers is a dense $G_\delta$-subset of $\mathbb{R}$;
\item for cardinality $\mathfrak{c}$ members of $\mathcal{L}_\varepsilon$, $x^x$ are Liouville numbers, and an equal number are not Liouville numbers.
\end{enumerate}
 We ask: what can be said about the set all Liouville numbers $x$ such that $x^x$ is transcendental, or indeed a Liouville number? The theorems below shed some light on this question.

\begin{proof}
The proof relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}. We refer the reader to \cite[Proposition 3.2]{MarquesOliveira2023} for details.
\end{proof}

In this article, we introduce the set of tuned spiffy constants, a proper  subset of cardinality $\mathfrak{c}$ of $\varepsilon$-strong Liouville numbers, with the property that if $x$ belongs to this set, then $x^x$ is a Liouville number.

\begin{remark}
Throughout this work, we employ standard asymptotic notation. For functions $f$ and $g$, we write $f(x) = O(g(x))$ or equivalently $f(x) \ll g(x)$ if $|f(x)| \le C|g(x)|$ for an absolute constant $C>0$ and all sufficiently large $x$. We denote a negligible error by $f(x) = o(g(x))$, which means $f(x)/g(x) \to 0$ as $x \to \infty$, and $f(x) \asymp g(x)$ signifies that $f\ll g$ and $g\ll f$. Unless stated otherwise, all implied constants are absolute.
\end{remark}

The proof of Theorem~\ref{thm:trans} \emph{relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}}. (Details are deferred to \cite[Proposition 3.2]{MarquesOliveira2023}.)

In establishing the main theorem, the proof \emph{relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}}. (Details are deferred to \cite[Proposition 3.2]{MarquesOliveira2023}.)

Question:
Let $x$ be a real number such that for some $varepsilon>0$ there exists an infinite sequence of rational numbers $p_k/q_k$ with $q_k>1$ satisfying
\[
\left|x-\frac{p_k}{q_k}\right|<q_k^{-(\log q_k)^{1+\varepsilon}}\qquad \text{for all }k\ge 1.
\]
Such a number is called an $\varepsilon$-strong Liouville number, and the set of all such numbers is denoted $\mathcal L_\varepsilon$. Under these assumptions, which statement about $x^x$ holds?

$x^x$ is transcendental.

Choices:
B. $x^x$ is a Liouville number.
C. $x^x$ is irrational.
D.
There exists a constant $\tau>0$, depending only on $\varepsilon$, such that for every $x\in\mathcal L_\varepsilon$ and all but finitely many rationals $a/b$ one has
\[
\left|x^x-\frac ab\right|\ge b^{-\tau}.
\]
E. If $x\in\mathcal L_\varepsilon$ for every $\varepsilon>0$, then $x^x$ is transcendental; but from membership in a single class $\mathcal L_\varepsilon$ for some fixed $\varepsilon>0$ one cannot conclude transcendence of $x^x$.


\label{marker}
\end{figure}
The categories at the right are pushouts, which are determined by the rest of diagram and therefore equivalent.
\end{proof}

\bibliographystyle{alpha}
\bibliography{refere

Mirror symmetry was first observed as an unexpected phenomenon in string theory in the late $1980$s. This phenomenon bridges two seemingly unrelated areas of mathematics: algebraic geometry and symplectic geometry. The resulting conjectures have since revolutionized both fields. In $1994$, at the ICM, Kontsevich \cite{Kon94} proposed that mirror symmetry reflects an equivalence of two categories
\[
D^\pi \mathrm{Fuk} (X) \cong D^b \mathrm{Coh}(\check{X}),
\]
between the derived Fukaya category of a Calabi--Yau manifold $X$ and the derived category of coherent sheaves on the mirror Calabi-Yau $\check{X}$. This formulation, known as homological mirror symmetry, has been generalized beyond Calabi-Yau manifolds. While verifying homological mirror symmetry for various examples is an active field, it also serves as a tool for transferring ideas and insights between symplectic and algebraic geometry.

A particularly timely direction in the study of homological mirror symmetry is to understand how Fukaya categories and the equivalence conjectured by Kontsevich behave under geometric decompositions of spaces into simpler building blocks. The setting in which this has been most successful so far are Liouville manifolds, which are non-compact symplectic manifolds with exact symplectic structure and cylindrical ends.
However, when Liouville manifolds have boundaries, defining Floer homology becomes challenging because the pseudoholomorphic curves (whose counts are used to define operations in Floer theory) might escape through the boundary. This leads to the notion of \emph{Liouville sector}, first introduced in a recent work of Ganatra-Pardon-Shende \cite{MR4106794}.

This paper explores the application of this technology to the study of symmetric products of open Riemann surfaces, which are smooth complex manifolds. It is a particularly important class of Liouville manifolds in light of applications to low-dimensional topology, such as Heegaard Floer homology by Ozsv\'ath and Szab\'o \cite{MR2113020}. Tim Perutz \cite{Per05} gave a careful description of a symplectic structure on the symmetric product. (The naive symplectic structure on the symmetric product induced by the symplectic form on the surface is not smooth on the diagonal.) Auroux \cite{MR2755992, MR2827825} provided a symplectic interpretation of bordered Heegaard-Floer homology through Fukaya categories of symmetric products. This application provides further motivation to study the symplectic geometry and the Fukaya categories of symmetric products.

We study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.

We study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$. However, it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.

\begin{theoremB}
For any $2$-dimensional Liouville sector $X$, the second symmetric product of $X$ is deformation equivalent to a Liouville sector $Y$. Moreover, the completion $\widehat{Y}$ is deformation equivalent to the second symmetric product $Sym^2(\widehat X)$ of the completion of $X$.
\end{theoremB}
Furthermore, we can apply the decomposition theorem (Theorem A) to homological mirror symmetry. Namely, a sectorial decomposition of $\Sym^2(\Sigma)$ opens the perspective of constructing a mirror space of the symmetric product by finding mirrors to the various Liouville sectors into which it can be decomposed and gluing them to each other in a suitable manner.

The rest of this paper is organized as follows.
Section $2$ reviews the basic theory of Liouville sectors and related notions such as sectorial domains, completions and truncations.
Section $3$ develops a local model on the symmetric square of the complex plane while section $4$ establishes the existence of quadratic Stein structures on surfaces with embedded arcs, providing the analytic framework for our main construction.
Section $5$ uses this structure to construct a global sectorial decomposition of the symmetric product of a surface.
Section $6$ studies the fibers of the sectorial boundaries and their completions, showing their Liouville equivalences.
Finally, Section $7$ applies these results to the symmetric product of the four-punctured sphere and proves homological mirror symmetry for this example.

\begin{defn}
 Let $\Sigma $ be a Riemann surface and $\varphi$ be a proper plurisubharmonic function on $\Sigma$. Let $\{s_i\}_{i\in I}$ be the set of saddles of $\varphi$ and $\mathcal{N}(\gamma_i)$ be the tubular neighborhood of the stable manifold $\gamma_i$ of the saddle $s_i$.\\
 We say $(\Sigma, \varphi)$ is a Riemann surface with a quadratic Stein structure if $\varphi |_{\mathcal{N}(\gamma_i)}$ is quadratic in local coordinates.
\end{defn}

\begin{proposition}
\label{global}
 For any non compact topological surface $\Sigma$ with disjoint proper embedded arcs $\{\gamma_i \}, i \in I$,
 we can build a quadratic Stein structure with one saddle $s_i$ (and possibly other saddles as well)
 and on each $\gamma_i$ and one minimum $m_j$
 on each component of $\Sigma - \bigcup \gamma_i$.
\end{proposition}

\section{Construction of sectorial decomposition}
\subsection{Statement and notations}
\begin{theorem} \label{mainthm}
Let $(\Sigma,\varphi)$ be a Riemann surface with a quadratic Stein structure and let $\{s_i\}$ the set of saddles and $\{m_j\}$ the set of minimum. This determines a sectorial collection of hypersurfaces $H_{s_i}$ in $Sym^2(\Sigma)$, which decompose $Sym^2(\Sigma)$ into a union of sectors with corners $U_{m_i,m_j}$.
\end{theorem}

\begin{corollary}
 Let $m$ be the number of arcs and $n$ be the number of the connected components of $\Sigma - \cup_{i\in I} \gamma_i$.
 The decomposition constructed consists of $\binom{n+1}{2}$ Liouville sectors with corners separated by $mn$ pieces of smooth hypersurfaces that meet transversely at $\binom{m}{2}$ corners.
 \end{corollary}

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$. However, it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.

In general, this theorem states that given a surface's sectorial decomposition, we can construct a sectorial decomposition of its second symmetric product.

The proof of theorem A relies on two main ingredients. One is the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface. The other ingredient is the
calculation on a local model described in section $3$. The methods used to prove Theorem A also imply other results for sectors and symmetric products of Riemann surfaces. For example,

\begin{theoremB}
For any $2$-dimensional Liouville sector $X$, the second symmetric product of $X$ is deformation equivalent to a Liouville sector $Y$. Moreover, the completion $\widehat{Y}$ is deformation equivalent to the second symmetric product $Sym^2(\widehat X)$ of the completion of $X$.
\end{theoremB}
Furthermore, we can apply the decomposition theorem (Theorem A) to homological mirror symmetry. Namely, a sectorial decomposition of $\Sym^2(\Sigma)$ opens the perspective of constructing a mirror space of the symmetric product by finding mirrors to the various Liouville sectors into which it can be decomposed and gluing them to each other in a suitable manner.

Ganatra, Pardon, and Shende \cite{MR4106794, MR4695507} have introduced and studied  wrapped Fukaya categories of Liouville sectors. The upshot of their work is that the wrapped Fukaya category of a Liouville manifold can be determined from a decomposition into Liouville sectors by a local-to-global principle.

In section $7$, we discuss the example of the symmetric product of the $4$-punctured sphere $\Sym^2(\mathbb{P}^1 - 4 \mbox{pts})$, which is also known as the \emph{complex $2$-dimensional pair of pants}. Lekili and Polishchuk \cite{MR4120165} have established homological mirror symmetry for this example using a computation of Auslander algebras. A geometric interpretation of mirror symmetry can be deduced from our sectorial decomposition results. In this case, the local-to-global principle in  \cite{MR4695507} yields the top diamond of the commutative diagram shown in Figure \ref{marker},
which describes the wrapped Fukaya category of $\text{Sym}^2(\mathbb{P}^1 - 4\text{pts})$ as the pushout of a diagram of wrapped Fukaya
categories of simpler Liouville sectors. Homological Mirror Symmetry for the $2$-dimensional pair of pants follows by identifying the mirror of each Liouville sector and the gluing maps between them.

The rest of this paper is organized as follows.
Section $2$ reviews the basic theory of Liouville sectors and related notions such as sectorial domains, completions and truncations.
Section $3$ develops a local model on the symmetric square of the complex plane while section $4$ establishes the existence of quadratic Stein structures on surfaces with embedded arcs, providing the analytic framework for our main construction.
Section $5$ uses this structure to construct a global sectorial decomposition of the symmetric product of a surface.
Section $6$ studies the fibers of the sectorial boundaries and their completions, showing their Liouville equivalences.
Finally, Section $7$ applies these results to the symmetric product of the four-punctured sphere and proves homological mirror symmetry for this example.

\paragraph{Acknowledgement.}
I would like to thank my PhD advisor Denis Auroux for his guidance, support, and generosity, as well as for the many ideas he contributed to this work. I am also grateful to John Pardon for helpful conversations and suggestions. This work was partially supported by NSF grant DMS-2202984, by the Simons Foundation (grant \#$385573$, Simons Collaboration on Homological Mirror Symmetry), and by the Simons Collaboration on New Structures in Low-Dimensional Topology.

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$, but “it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.”

“The proof of theorem A relies on two main ingredients.” First, “the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface.” Second, “the calculation on a local model described in section $3$.”

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$, but “it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.”

“The proof of theorem A relies on two main ingredients.” First, “the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface.” Second, “the calculation on a local model described later.”,

Let $\Sigma$ be a topological surface with disjoint proper embedded arcs $\{\gamma_i\}_{i\in I}$. For suitably defined Stein structures on $\Sigma$ and $\Sym^2(\Sigma)$, the arcs $\{\gamma_i\}_{i\in I}$ determines a sectorial collection of smooth hypersurfaces $H_{_i}$ in $\Sym^2(\Sigma)$ which separates the symmetric product into Liouville sectors.,

Question: Let $\Sigma$ be a topological surface equipped with a collection of pairwise disjoint proper embedded arcs $\{\gamma_i\}_{i\in I}$. Assume one has suitably defined Stein structures on $\Sigma$ and on its second symmetric product $\operatorname{Sym}^2(\Sigma)=(\Sigma\times\Sigma)/\mathfrak{S}_2$. Under these hypotheses, which conclusion about $\operatorname{Sym}^2(\Sigma)$ holds?

The arcs $\{\gamma_i\}_{i\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ in $\operatorname{Sym}^2(\Sigma)$, and these hypersurfaces separate $\operatorname{Sym}^2(\Sigma)$ into Liouville sectors.

Choices:
B. The arcs $\{\gamma_i\}_{i\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ in $\operatorname{Sym}^2(\Sigma)$, and these hypersurfaces separate $\operatorname{Sym}^2(\Sigma)$ into Weinstein domains with boundary.
C. The arcs $\{\gamma_i\}_{i\in I}$ determine a collection of smooth hypersurfaces $H_i$ in $\operatorname{Sym}^2(\Sigma)$ which decompose $\operatorname{Sym}^2(\Sigma)$ into pieces compatible with the given Stein structures.
D. The arcs $\{\gamma_i\}_{i\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ directly by taking configurations in which one of the points lies on $\gamma_i$, and this immediately separates $\operatorname{Sym}^2(\Sigma)$ into Liouville sectors.
E. For any suitably defined Stein structures on $\Sigma$ and $\operatorname{Sym}^2(\Sigma)$, there is a single Liouville sector $Y$ deformation equivalent to $\operatorname{Sym}^2(\Sigma)$ whose boundary is the union of the hypersurfaces $H_i$ determined by the arcs $\{\gamma_i\}_{i\in I}$.

target pieces are Liouville sectors, not Weinstein domains

drops the sectorial and Liouville-sector conclusions

ignores the singular-corner obstruction to the naive configuration-space construction

replaces a decomposition into multiple sectors by a single deformation-equivalent Liouville sector


\begin{theorem}\label{main thm 1}
Let $r\ge 2$ be an integer and $\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \ref{Table 1}. Let $a(r) = 2\log{2}\left(3.0758r+1.38402\log(4r)-1.5379\right)$. Then, for $q \ge \max\{10^{1145},e^{e^{a(r)}}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$

    Furthermore, as $q\rightarrow\infty$, we have a constant $D(r)$ from Table \ref{Table 1} such that
    $$|S_{\chi}(M,N)| \le (D(r)+o(1))N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

Let $q$ be a positive integer.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and~$N\ge1$, we consider the character sum
\begin{equation*}
 S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}

Burgess, in a series of papers~\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\left|S_{\chi}(M,N)\right| \ll N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+\eps},$$
for $r = 2,3$ and for any $r\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\chi}(M,N)$. In the case when $q$ is prime, Booker \cite{Boo2006} proved a result for $\chi$ quadratic, and McGown \cite{mcgown2012} and Trevi\~no \cite{ET2} made improvements for general $\chi$. The current best result for $q$ prime and $2\le r\le 10$ is due to Francis \cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \cite{ET2})
$$|S_{\chi}(M,N)| \le 2.74 N^{1-\frac{1}{r}}p^{\frac{r+1}{4r^2}}(\log{p})^{\frac{1}{r}},$$
for all integers $r\ge 2$ and for all primes $p\ge 10^7$.

These explicit results have been used to determine the class number of a quadratic field of large discriminant \cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \cite{mcgown2012}, to bound the least primitive root modulo a prime \cite{COT-2016}, to give an explicit bound for $L(1,\chi)$ in the case of quadratic characters \cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \cite{Elkies}, among other applications.

In the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \cite{JKL}.
\begin{theoremA*}
 Let $\chi$ be a primitive character mod $q$ with $q \ge e^{e^{9.594}}$. Then, for $N\le q^{\frac{5}{8}}$,
 $$\left|S_{\chi}(M,N)\right|\le 9.07 \sqrt{N} q^{\frac{3}{16}} \log^{\frac{1}{4}}(q)\left(2^{\omega(q)}\tau(q)\right)^{\frac{3}{4}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{2}},$$
 where $\tau(q)$ is the number of divisors of $q$ and $\omega(q)$ is the number of distinct prime factors of~$q$.
\end{theoremA*}

In other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\ge 3$ when $q$ is cubefree.

To state our theorem, we use the following standard labels of arithmetic functions:
let $\omega(q)$ denote the number of distinct prime factors of~$q$, let $\phi(q)$ be Euler's totient function, and let $\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\tau(n) = \tau_2(n)$ is the ordinary number-of-divisors function. Also define
\begin{equation} \label{meet Mr. Q}
m_r(q) = \min \biggl\{ \tau_{2r}(q), \biggl( \frac{\tau(q)}2 \biggr)^{2r-1} , \frac{q}{2r} \biggr\}.
\end{equation}

\label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{

To state our theorem, we use the following standard labels of arithmetic functions:
let $\omega(q)$ denote the number of distinct prime factors of~$q$, let $\phi(q)$ be Euler's totient function, and let $\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\tau(n) = \tau_2(n)$ is the ordinary number-of-divisors function. Also define
\begin{equation} \label{meet Mr. Q}
m_r(q) = \min \biggl\{ \tau_{2r}(q), \biggl( \frac{\tau(q)}2 \biggr)^{2r-1} , \frac{q}{2r} \biggr\}.
\end{equation}

\begin{table}[h!]
\begin{center}
 \begin{tabular}{ | c | c | c|}
 \hline
 $r$ & $C(r)$ & $D(r)$ \\ \hline
 2 & 15.219 & 8.362 \\ \hline
 3 & 5.359 & 4.581\\ \hline
 4 & 3.671 & 3.396\\ \hline
 5 & 2.953 & 2.811\\ \hline
 6 & 2.549 & 2.462\\ \hline
 7 & 2.290 & 2.229\\ \hline
 8 & 2.108 & 2.063\\ \hline
 9 & 1.973 & 1.938\\ \hline
 $\ge$10 & 1.869 & 1.841 \\ \hline
 \end{tabular}
 \caption{Constants in the Burgess inequality for values of $r$.}\label{Table 1}
\end{center}
\end{table}

\begin{theorem}\label{main theorem 2}
 Let $\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \ref{Table 1}. Then, for $q \ge \max\{10^{1145}, 2^{4r-2}\}$, if $r=2$ or $q$ is cubefree, we have
 $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

Let $r\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\ge 3$ for the remainder of the paper.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and ~$N\ge1$, we consider the character sum
\begin{equation*}
 S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}
Given an integer $r\ge2$, we use induction to prove a bound of the form
\begin{equation} \label{eq desired bound}
| S_\chi(M,N) | \le E_{q,r}(N), \quad\text{where }
E_{q,r}(N) = C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q),
\end{equation}
where $C\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \eqref{meet Mr. Q}, and
\begin{equation}\label{meet T(q)}
T(q) = \begin{cases}\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}-\frac{1}{2r^2}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main thm 1},\\
\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main theorem 2}
.\end{cases}
\end{equation}

\begin{lemma}\label{lem: alpha bound}
Let $r,q\ge 2$ and $N \ge q^{\frac{1}{4}+\frac{1}{4r}} $ be integers and let $i \in \{1,2\}$. Let $A = A_i(r,q)$, $B = B_i(r,q)$, $\kappa$ be defined as in \eqref{choice of kappa}, and $t_i(r)$ be defined as in Lemma \ref{A lower bound lemma}.
Let
\begin{equation*}
 \alpha
 =
 \left[ \frac{\left(\kappa\left(\frac{\phi(q)}{q}\right)+\frac{B2^{\omega(q)-1}}{N} \right)^2}{B}
 +
 2\kappa\left(\frac{6}{\pi^2}\log{(A)}+\delta+\frac{2\log{(A)}+2}{A-1}\right)\right],
\end{equation*}
where $\delta = \frac6{\pi^2} - \frac{36\zeta'(2)}{\pi^4} \approx 0.954422$.
Then, for $q\ge \max\{10^{1145}, (10r)^{18}, t_i(q)\}$,
$$\alpha \le \frac{4}{3}\kappa \log{q}.$$
\end{lemma}

\begin{lemma}\label{lemma to set up the main theorems}
For $i\in\{1, 2\}$ and $T_i$ be defined as in \eqref{eq T1} and \eqref{eq T2}. Let $s, \alpha, u, $ and $w$ be defined as in \eqref{eq s}, \eqref{eq alpha}, \eqref{eq u}, and \eqref{w}, respectively. Then
 \begin{equation*}
\begin{split}
 \left| S_\chi(M, N) \right|
 \le
 \left(\frac{B}{B-1}\right)N^{1-1/r}q^{(r+1)/{4r^2}}(\log q)^{\frac{1}{2r}}
 \left(
 u(\alpha w)^{\frac{1}{2r}}
 +sT_i(q)
 \right).
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
 Fix $i\in \{1,2\}$ and let $q\ge \max\{10^{1145},t_i(r)\}$. From Lemma \ref{eq after Holder}, once we choose $A$ and $B$ as described in Definition \ref{def A B v}, replacing $T(q)$ with $T_i(q)$ and $C(r)$ with $C$, we have
\begin{align*}
 \left| S_\chi\left(M, N\right)\right|
 &\le
 \frac{1}{\#\cA\cdot \#\cB}\left(\sum_{x=1}^q v_\cA(x)\right)^{1-\frac{1}{r}}
 \left(\sum_{x=1}^q v^2_{\cA}(x)\right)^{\frac{1}{2r}}\left(\sum_{x=1}^q\left|\sum_{b \in \cB}\chi\left(x+b\right)\right|^{2r}\right)^\frac{1}{2r}
 \\
 &\qquad{}+
\frac{2C}{2-1/r} A^{1-1/r} \frac{(\lfloor B\rfloor+1)^{2-1/r}}{\#\cB} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T_i(q). \end{align*}

Similar to the proof of Theorem \ref{main thm 1}, $10^{1145}\ge (10r)^{18}$ for $r\le 10^{50}$ while $t_2(r) \ge (10r)^{18}$ for $r\ge 10^{50}$, thus from \eqref{roughly u} and \eqref{eq alpha upper bound},
\begin{equation*}
 u \le \frac{2^{1/r}}{q^{\frac{1}{2r^2}}}\left(\frac{N}{A\phi^*}\right)^{1/r},
\end{equation*}
and
\[
\alpha \le \frac{4}{3}\kappa \log{q}.
\]
Furthermore,
\begin{equation}\label{f2}
f_2 = (4r)^{\omega(q)}m_r(q)(r-1)+1 \le r(4r)^{\omega(q)}m_r(q),
\end{equation}
and \begin{equation}\label{beta B def 2}\beta = \left(\frac{r-1}{(2r)r!}\right)^{\frac{1}{r}},
\end{equation}
therefore we bound $w$ in \eqref{w} as
\begin{equation}\label{bounding w 2}
 w \le \frac{2^{1+1/r}(r\cdot r!)^{1/r}}{(r-1)^{1+1/r}\log q} \left(4r\right)^{\omega(q)}m_r(q).
\end{equation}
Using \eqref{beta B def 2}, $B = r^2q^{\frac{1}{2r}} \beta$, and $AB = \kappa N$, we get
\begin{equation}\label{bounding N over A 2}
\frac{N}{A q^{\frac{1}{2r}}} = \frac{r^2}{\kappa}\left(\frac{r-1}{r!(2r)}\right)^{\frac{1}{r}}.
\end{equation}
Therefore, combining \eqref{eq s}, \eqref{roughly u}, \eqref{eq alpha upper bound}, \eqref{bounding w 2}, and \eqref{bounding N over A 2}, we get
\begin{align*}
 &u(\alpha w)^{\frac{1}{2r}}+sT_2(q)\\
 &\le 2^{\frac{1}{r}}\left(\frac{N}{Aq^{\frac{1}{2r}}\phi^*}\right)^{\frac{1}{r}}\left(\frac{4}{3}\kappa \log{q}\right)^{\frac{1}{2r}}\left(\frac{2^{1+\frac{1}{r}}(r\cdot r!)^{\frac{1}{r}}r^{\frac{1}{r}}}{(r-1)^{1+\frac{1}{r}}\log{q}}\right)^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\\ &+\frac{2C\kappa^{1-\frac{1}{r}}}{2-\frac{1}{r}}\left(\frac{B+1}{B}\right)^{2-\frac{1}{r}}T_2(q)\\
 &= \left(2^{\frac{5}{2r}-\frac{1}{2r^2}}3^{-\frac{1}{2r}}(r!)^{-\frac{1}{2r^2}}(r-1)^{\frac{1}{2r^2}-\frac{1}{2r}}r^{\frac{2}{r}-\frac{1}{2r^2}}\kappa^{-\frac{1}{2r}} +\frac{2C\kappa^{1-\frac{1}{r}}}{2-\frac{1}{r}}\left(\frac{B+1}{B}\right)^{2-\frac{1}{r}}\right)T_2(q).
\end{align*}
This produces the same conditions on $C$ and $D$ as in the proof of Theorem \ref{main thm 1}, thus we obtain the constants in Table \ref{Table 1}.
\end{proof}

\begin{table}[h!]
\begin{center}
  \begin{tabular}{ | c | c | c|}
    \hline
    $r$ & $C(r)$ & $D(r)$ \\ \hline
    2 & 15.219 & 8.362 \\ \hline
    3 & 5.359 & 4.581\\ \hline
    4 & 3.671 & 3.396\\ \hline
    5 & 2.953 & 2.811\\ \hline
    6 & 2.549 & 2.462\\ \hline
    7 & 2.290 & 2.229\\ \hline
    8 & 2.108 & 2.063\\ \hline
    9 & 1.973 & 1.938\\ \hline
    $\ge$10 & 1.869 & 1.841 \\ \hline
  \end{tabular}
  \caption{Constants in the Burgess inequality for values of $r$.}\label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{theorem}\label{main theorem 2}
    Let $\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \ref{Table 1}. Then, for $q \ge \max\{10^{1145}, 2^{4r-2}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

To prove Theorems \ref{main thm 1} and \ref{main theorem 2} we need the following explicit Weil-type inequality, which we consider of independent interest.

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

The paper is organized as follows. We prove Theorem \ref{weil} in Section~\ref{sec: weil}. In Section~\ref{General setup} we describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \cite[Theorem 12.6]{IK2004}. We prove other technical lemmas in Section~\ref{sec: technical}, some important bounds in Section~\ref{section asymptotics}, and we prove our main theorems in Section~\ref{sec: main theorems}.

\subsection*{Notation}
Most of the notation in the paper is standard, for example, $\phi(n)$ is Euler's totient function, for a finite set $S$, $\#S$ is the number of elements of $S$, $\lfloor x\rfloor$ is the floor of $x$, for a pair of integers $a,b$, $(a,b) = \gcd(a,b)$. We also use Landau's $O$ notation, i.e., $f(x) = O(g(x))$ if there exists a positive constant $C$ such that $|f(x)| \le C|g(x)|$ for large enough $x$. A notation we use that is less well-known is $O^*$. We say that $f(x) = O^*(g(x))$ if for all $x\ge x_0$, $|f(x)| \le |g(x)|$.

To prove Theorems \ref{main thm 1} and \ref{main theorem 2} the authors say they "need the following explicit Weil-type inequality" (Theorem \ref{weil}). They then describe the structure of the proof: they prove Theorem \ref{weil} in Section~\ref{sec: weil}, in Section~\ref{General setup} they "describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \cite[Theorem 12.6]{IK2004}," they prove "other technical lemmas" in Section~\ref{sec: technical} and obtain "some important bounds" in Section~\ref{section asymptotics}, and finally they "prove our main theorems" in Section~\ref{sec: main theorems}.

To prove the main theorem and Theorem \ref{main theorem 2} the authors say they “need the following explicit Weil-type inequality”. We first record this Weil-type inequality.

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

They then describe the structure of the proof as follows: they prove Theorem \ref{weil} earlier in the paper; next, in

\begin{align}\label{eq bad tuples}
    \sum_{(b_1,b_2,\ldots,b_{2r})\in \cB_2}\sum_{x\bmod{q}}\chi(f_1(x)f_2^{(\phi(q)-1)}(x)) \le r^{2r} \binom{\lfloor B\rfloor}{r} q \le \frac{r^{2r} \lfloor B\rfloor^r}{r!}q.
\end{align}
Combining the bounds in \eqref{eq good tuples} and \eqref{eq bad tuples}, we get the desired upper bound in the Weil-type inequality.
\end{proof}

\section{General setup}\label{General setup}

Let $r\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\ge 3$ for the remainder of the paper.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and ~$N\ge1$, we consider the character sum
\begin{equation*}
    S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}
Given an integer $r\ge2$, we use induction to prove a bound of the form
\begin{equation} \label{eq desired bound}
| S_\chi(M,N) | \le E_{q,r}(N), \quad\text{where }
E_{q,r}(N) = C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q),
\end{equation}
where $C\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \eqref{meet Mr. Q}, and
\begin{equation}\label{meet T(q)}
T(q) = \begin{cases}\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}-\frac{1}{2r^2}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main thm 1},\\
\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main theorem 2}
.\end{cases}
\end{equation}

We take $1\le N\le q^{\frac{1}{4}+\frac{1}{4r}}$ as the base case. In this case, the inequality~\eqref{eq desired bound} follows from the trivial bound $|S_\chi(M,N)| \le N$, since
\begin{align*}
&N \le C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q) \\
\implies &N^{1/r} \le C q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q) \\
\implies &N \le q^{\frac{r+1}{4r}} \bigl( C (\log q)^{\frac{1}{2r}} T(q) \bigr)^r,
\end{align*}
when $N \le q^{\frac{1}{4}+\frac{1}{4r}}$ (because $C(\log q)^{\frac{1}{2r}} T(q) \ge 1$).
\\
Therefore, in the induction step, we assume
\begin{align}\label{inductive hypothesis}
\text{for $N > q^{\frac{1}{4}+\frac{1}{4r}}$, the bound in \eqref{eq desired bound} holds for all smaller positive integers.}
\end{align}

they “describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in E. Kowalski and H. Iwaniec, \cite[Theorem 12.6]{IK2004}.” They then prove other technical lemmas later, obtain some important bounds later, and finally establish the main theorem together with Theorem \ref{main theorem 2}.

\label{main thm 1}
Let $r\ge 2$ be an integer and $\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{. Let $a(r) = 2\log{2}\left(3.0758r+1.38402\log(4r)-1.5379\right)$. Then, for $q \ge \max\{10^{1145},e^{e^{a(r)}}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$

    Furthermore, as $q\rightarrow\infty$, we have a constant $D(r)$ from Table \label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{ such that
    $$|S_{\chi}(M,N)| \le (D(r)+o(1))N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$,

Question:
Let $q$ be a positive integer, let $M$ be an integer and $N\ge 1$, and for a primitive Dirichlet character $\chi \pmod q$ define
\[
S_\chi(M,N)=\sum_{M<n\le M+N}\chi(n).
\]
For arithmetic functions, let $\omega(q)$ be the number of distinct prime factors of $q$, let $\phi(q)$ be Euler’s totient function, let $\tau_k(q)$ be the number of ordered factorizations of $q$ into $k$ positive integers (so $\tau(q)=\tau_2(q)$), and define
\[
m_r(q)=\min\!\left\{\tau_{2r}(q),\left(\frac{\tau(q)}{2}\right)^{2r-1},\frac{q}{2r}\right\}.
\]
Fix an integer $r\ge 2$, and set
\[
a(r)=2\log 2\,\bigl(3.0758r+1.38402\log(4r)-1.5379\bigr).
\]
Assume that $\chi$ is primitive modulo $q$, that
\[
q\ge \max\{10^{1145},e^{e^{a(r)}}\},
\]
and that either $r=2$ or $q$ is cubefree. Let $C(r)$ and $D(r)$ be given by
\[
\begin{array}{c|ccccccccc}
r&2&3&4&5&6&7&8&9&\ge 10\\ \hline
C(r)&15.219&5.359&3.671&2.953&2.549&2.290&2.108&1.973&1.869\\
D(r)&8.362&4.581&3.396&2.811&2.462&2.229&2.063&1.938&1.841
\end{array}
\]
Which of the following conclusions about $S_\chi(M,N)$ is valid under these hypotheses?

For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
Moreover, as $q\to\infty$, one has the asymptotic refinement
\[
|S_{\chi}(M,N)|\le (D(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]

Choices:
B.
For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
Moreover, as $q\to\infty$, one has the asymptotic refinement
\[
|S_{\chi}(M,N)|\le (D(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
C.
For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
D.
For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r},
\]
provided $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and $q$ is cubefree.
Moreover, as $q\to\infty$, one has the asymptotic refinement
\[
|S_{\chi}(M,N)|\le (D(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
E.
There exists a constant $C(r)$ depending only on $r$ such that for every primitive Dirichlet character $\chi\pmod q$ with $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, one has for every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r},
\]
and moreover the same asymptotic statement holds with $D(r)$ replaced by $C(r)$:
\[
|S_{\chi}(M,N)|\le (C(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]

exponent of ((4r)^{\omega(q)}m_r(q)) in main theorem vs weaker theorem

dropped the asymptotic refinement involving D(r)+o(1)

special exception allowing r=2 without cubefree q

replacement of the sharper asymptotic constant D(r) by C(r)