\begin{theorem}\label{thm:main-dem}
	Let $\eps \in (0,2^{-10})$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
	Let $G$ be an $n$-vertex $(\eps,p)$-typical graph and $k = \eps 2^{-10}p^{-1} \log pn$.
	Then, the following holds for all sufficiently large $n$. With probability at least $1- n^{-2^{-11}\log pn}$, in the random greedy independent set process, we have
	\[ d_i(v) = (1 \pm f_i)\td_i\]
	for all $v \in V_i$ and all $i \in [k-1]$.
\end{theorem}

\begin{theorem}\label{thm:main-prague}
	Let\footnote{In all our statements, $\eps$ is a fixed constant and $n$ tends to infinity.} $\eps \in (0,1)$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
	Then, with high probability we have
	\begin{align*}
		\mathrm{pdim}(G_{n,p}) = O_{\eps} \left ( \dfrac{pn \log n}{\log pn} \right ).
	\end{align*}
\end{theorem}

Motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem~\cite{Deuber,ErdosHajnalPosa,Rodl},
Nešetřil, Pultr and Rödl~\cite{Nesetril1977-pj,Nesetril1978-km} introduced the notion of the Prague dimension of a graph in the 1970s.
For a graph $G$, the Prague dimension $\text{pdim}(G)$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph.
This is closely related to $\text{idim}(G)$, defined as the minimum number of partial clique factors whose union equals $\overline{G}$, the complement of $G$.
In fact,
Nešetřil and Rödl~\cite{Nesetril1978-km} showed that for every graph $G$, we have
\begin{align}\label{eq:pdim-idim}
 \mathrm{idim}(G) \le \mathrm{pdim}(G) \le \mathrm{idim}(G) + 1.
\end{align}

In 1977, Lovász, Nešetřil, and Pultr~\cite{Lovasz1980-kw} showed that a graph on $n\ge2$ vertices has Prague dimension at most $n-1$ and classified all graphs which achieve this bound.
General upper and lower bounds were obtained by Alon~\cite{Alon1986-oc}, who proved that if $G$ is a graph on $n$ vertices with minimum degree at least 1 and maximum degree $\Delta$, then $\log (n/ \Delta)/ \log 2 \le \text{pdim}(G) = O(\Delta^2\log n)$; all our asymptotic expressions are meant as $n \to \infty$, and all logarithms are natural.
The upper bound was later improved by Eaton and Rödl~\cite{Eaton1996-ap}, who showed that
\begin{align*}
 \text{pdim}(G) = O(\Delta \log n).
\end{align*}
Moreover, they proved that this is best possible up to a factor of $1/(\log \Delta + \log \log (n/2\Delta) )$.
When $\Delta$ is a constant, these results provide the order of magnitude of $\text{pdim}(G)$.
Precise asymptotics are known for matchings, trees and cycles~\cite{Alon1986-oc,Lovasz1980-kw,Poljak1981-lt}, hypercubes~\cite{Krivka} and Kneser graphs $K(n,k)$ for constant values of $k$~\cite{Poljak1978-um}.
For more details, we refer the reader to~\cite{Kantor}.

Nešetřil and Rödl~\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\Omega_p(n/\log n)$ with high probability for constant values of $p$.
Many years later, Guo, Patton and Warnke~\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\cite{Pippenger1989-ak} type
edge-colouring result for random hypergraphs with large uniformities.
More recently, Molnar, Rödl, Sales and Schacht~\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.
Via a double counting argument, they showed that for $p \gg n^{-2}$, we have\footnote{For any two functions $f$ and $g$, we write $f \ll g$ to mean $f = o(g)$.}
\begin{align}\label{eq:lower-bound-MRSS}
 \text{pdim}(G_{n,p}) \ge \dfrac{pn \log \frac{1}{p} }{5\log n }
\end{align}
with high probability.
Furthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\varepsilon > 0$ and $p$ such that $n^{-1/3} \log^{4/3} n \ll p \ll n^{-\varepsilon}$, we have $\text{pdim}(G_{n,p}) = \Theta_{\varepsilon}(pn)$ with high probability.
In our first main result, we prove an upper bound that is valid for almost the entire range of $p$.

Nešetřil and Rödl~\cite{Nesetril1983-ju} pioneered the study of the Prague dimension of random graphs by showing that the Prague dimension of $G_{n,p}$ is $\Omega_p(n/\log n)$ with high probability for constant values of $p$.
Many years later, Guo, Patton and Warnke~\cite{Guo2023-wq} managed to obtain a matching upper bound by using a Pippenger--Spencer~\cite{Pippenger1989-ak} type
edge-colouring result for random hypergraphs with large uniformities.
More recently, Molnar, Rödl, Sales and Schacht~\cite{MolnarRodlSalesSchacht} studied the Prague dimension of sparse random graphs.
Via a double counting argument, they showed that for $p \gg n^{-2}$, we have\footnote{For any two functions $f$ and $g$, we write $f \ll g$ to mean $f = o(g)$.}
\begin{align}\label{eq:lower-bound-MRSS}
 \text{pdim}(G_{n,p}) \ge \dfrac{pn \log \frac{1}{p} }{5\log n }
\end{align}
with high probability.
Furthermore, by analysing the distribution of independent sets in~$G_{n,p}$, they showed that for all $\eps > 0$ and $p$ such that $n^{-1/3} \log^{4/3} n \ll p \ll n^{-\eps}$, we have $\text{pdim}(G_{n,p}) = \Theta_{\eps}(pn)$ with high probability.
In our first main result, we prove an upper bound that is valid for almost the entire range of $p$.

In particular, by combining our result with the lower bound in~\eqref{eq:lower-bound-MRSS}, we obtain that for all $\eps > 0$ and $p$ such that $ n^{-1+\eps} \le p \le n^{-\eps}$, with high probability the Prague dimension of $G_{n,p}$ is $\Theta_{\eps}(pn)$.
As it is crucial for our analysis to have concentration on the degrees of all vertices in $G_{n,p}$, we note that $n^{-1}\log n$ is a natural barrier for the range of $p$ our method can handle.

Our methods also allow us to bound the minimum number of independent sets needed to cover all the non-edges of $G=G_{n,p}$, denoted by $\overline{\theta}_1(G)$.
Initially, one may conjecture that with high probability $\overline{\theta}_1(G)$ should be of order $n^2/\alpha(G)^2$ for all values of $p$, which corresponds to the size of a `near' optimal clique cover of $\overline{G}$.
Frieze and Reed~\cite{Frieze1995-xa}, by improving the results of Bollobás, Erd\H{o}s, Spencer, and West~\cite{Bollobas1993-nl}, showed that this is indeed the case for constant values of $p$.
Surprisingly, this is not true for the sparse range.
In~\cite{Guo2023-wq}, Guo, Patton and Warnke proved that for any fixed $\eps >0$ and $n^{-1} \ll p \le 1-\eps$, with high probability we have
\begin{align*}
 \overline{\theta}_1(G_{n,p}) = \Omega \left ( \dfrac{n^2 \log \frac{1}{p} }{\alpha^2(G_{n,p})} \right ).
\end{align*}
It is worth mentioning that due to the work of Matula~\cite{Matula1970-ka}, Bollobás and Erd\H{o}s~\cite{Bollobas1976-al}, Grimmett and McDiarmid~\cite{Grimmett1975-vs}, and Frieze~\cite{Frieze1990-cu}, we know that the size of the largest independent set in $G_{n,p}$ is of order $\Theta(p^{-1}\log pn)$ for $n^{-1} \ll p \le 1- \eps$, for any fixed $\eps > 0$.
Also note that
$\log \frac{1}{p} = \Theta_{\eps}(\log n)$ for $ n^{-1}\ll p \ll n^{-\eps}$.

\begin{theorem}\label{thm:main}
 Let $\eps\in (0,1)$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
 Then, with high probability we have
 \begin{align*}
 \overline{\theta}_1(G_{n,p}) = \Theta_{\eps} \left ( \dfrac{n^2\log n}{\alpha^2(G_{n,p})} \right ).
 \end{align*}
\end{theorem}

\begin{theorem}\label{thm:prob-non-edge}
 Let $\eps \in (0,2^{-10})$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
 Let $G$ be an $n$-vertex $(\eps,p)$-typical graph and $k = \eps 2^{-10} p^{-1} \log pn $.
 Then, for all distinct $u, v \in V(G)$ such that $uv \notin E(G)$, we have
 \begin{align*}
 \Pr{v \in I_k} = \big(1+o_{\eps}(1) \big ) \dfrac{k}{n} \qquad
 \text{and} \qquad \Pr{u,v \in I_k} = \big (1+o_{\eps}(1) \big ) \left ( \dfrac{k}{n} \right )^2.
 \end{align*}
\end{theorem}

Let $\big(J_{i,j}: i \in [t], j \in [s] \big )$ be a collection of $ts$ independent copies of $I_k = I_k(G)$.
 For each $i \in [t]$, define $\mathcal{I}_i$ to be the collection of vertex-disjoint subsets of $V(G)$ given by
 \begin{align*}
 I_{i,j} \coloneqq J_{i,j} \setminus \big ( J_{i,1} \cup \cdots \cup J_{i,j-1} \big )
 \end{align*}
 for all $j \in [s]$.
 By~\eqref{eq:pdim-idim}, it suffices to show that with high probability every non-edge of $G$ is covered by some $I_{i,j}$.
 For $i \in [t]$, the probability that a given non-edge $uv$ is covered by an independent set in the partition $\mathcal{I}_i$ is equal to
 \begin{align}\label{eq:prob-non-edge}
 \sum_{j=1}^{s}\Pr{u,v \in I_{i,j}} & = \sum_{j=1}^{s} \Pr{u,v \in J_{i,j}} \prod_{\ell\le j-1} \Pr{ \{ u,v \} \cap J_{i,\ell} = \emptyset}\nonumber\\
 & = \sum_{j=1}^{s} \Pr{u,v \in I_k} \Pr{ \{ u,v \} \cap I_k = \emptyset}^{j-1}.
 \end{align}
 By the inclusion-exclusion principle, the probability that neither $u$ nor $v$ is covered by $I_k$ is
 \begin{align}\label{eq:prob-non-edge-2}
 \Pr{ \{ u,v \} \cap I_k = \emptyset} &= 1 - \Pr{u \in I_k} - \Pr{v \in I_k} + \Pr{u,v \in I_k} \nonumber \\
 & = 1 - \Theta_{\eps} \left ( \dfrac{k}{n} \right ) = 1 - \Theta_{\eps}(s^{-1}).
 \end{align}
 Here we use Theorem~\ref{thm:prob-non-edge} and the bound $k = o(n)$, which follows from the fact that $p \gg n^{-1}\log n$.

Let us also recall the definition of the error terms that we use to control the deviation of the actual degrees from the expected trajectory.
For $i \ge 0$, we set
\begin{align}\label{eq:error-function}
 f_0 \coloneqq 4\log n \sqrt{ \dfrac{\log pn}{pn} + p } \qquad \text{and} \qquad f_i := \left ( \dfrac{1+16p}{1-p} \right )^i f_0.
\end{align}
Note that the error is increasing, yet we have
\begin{align}\label{eq:bound-f}
 f_{i-1} < f_{i} \le f_k \le (pn)^{2^{-5}\eps} f_0 = o(1) \qquad \text{and} \qquad
 \dfrac{1}{1\pm f_i} = 1 \pm \dfrac{3}{2}f_i
\end{align}
for all $i \in [k]$ and $(\log n)^{2+\eps} n^{-1} \le p < n^{-\eps}$.
To derive the inequality $f_k \le (pn)^{2^{-5}\eps}f_0$, we simply use that $(1+16p)(1-p)^{-1} \le 1+2^5p \le e^{2^5p}$.
It follows from~\ref{item:P1} that
\begin{align}\label{eq:initial-parameters-2}
 |V_i| = (1\pm f_i)(1-p)^i n .
\end{align}
for all $i \in [k]$.
Other useful estimates are
\begin{align}\label{eq:asymptotics}
 (1-3p)^k \ge e^{-8pk} \ge (np)^{-2^{-7}\eps} \ge n^{-2^{-7}(\eps - \eps^2)} \ge n^{-2^{-7}\eps } \quad
 \text{and} \quad f_i > f_0 \ge n^{-1/2}
 \end{align}
for all $i \in [k]$.
Above we use that $1-x \ge e^{-2x}$ for all $x \in (0,1/2)$.

By Freedman's inequality, and using that $f_0 = o(1)$ and $\td_0 = pn$, we have
 \begin{align}\label{eq:freedman-appl-1}
 \Pr{ X_{v,i}^{-}-X_{v,0}^{-} \ge \dfrac{f_0\td_0}{2}} & \le
 \exp\left(-\frac{(f_0\td_0)^2}{2^{12}(p^2n+\log n)(pn + f_0\td_0)}\right) \nonumber \\
 & \le \exp\left(-\frac{f_0^2pn}{2^{13}(p^2n+\log n)}\right) \nonumber \\
 & \le \exp\left(-\frac{\log^2 n \log pn + p^2n \log^2 n }{2^{9}(p^2n+\log n)}\right).
 \end{align}
 In the last inequality, we use the definition of $f_0$ (see~\eqref{eq:error-function}).
 Let $E$ denote the absolute value of the exponent in the last term.
 If $p^2n \ge \log n$, then $E \ge 2^{-9}p^2n\log^2n(p^2n+\log n)^{-1} \ge 2^{-10}\log^2n$, and hence
 the right-hand side of~\eqref{eq:freedman-appl-1} is at most $\exp(-2^{-10}\log^2 n)$; if $p^2n < \log n$, then $E \ge 2^{-9}\log^2 n \log pn(p^2n+\log n)^{-1} \ge 2^{-10}\log n \log pn$, and hence
 the right-hand side of~\eqref{eq:freedman-appl-1} is at most $\exp(-2^{-10}\log n \log pn)$.
 Therefore, in both cases we have
 \begin{align*} \Pr{X_{v,i}^{-}-X_{v,0}^{-} \ge \dfrac{f_0\td_0}{2}} \le n^{-2^{-10}\log pn} \quad \text{and} \quad
 \Pr{-X_{v,i}^{+}+X_{v,0}^{+} \ge \dfrac{f_0\td_0}{2}} \le n^{-2^{-10}\log pn}.
 \end{align*}
 Summing over all $v \in V_0$ and $i \in [k-1]$, we obtain that $\Pr{\tau < k} \le n^{-2^{-11}\log pn}.$
\end{proof}

In particular, by combining our result with the lower bound in~\eqref{eq:lower-bound-MRSS}, we obtain that for all $\varepsilon > 0$ and $p$ such that $ n^{-1+\varepsilon} \le p \le n^{-\varepsilon}$, with high probability the Prague dimension of $G_{n,p}$ is $\Theta_{\varepsilon}(pn)$.
As it is crucial for our analysis to have concentration on the degrees of all vertices in $G_{n,p}$, we note that $n^{-1}\log n$ is a natural barrier for the range of $p$ our method can handle.

To prove Theorem~\ref{thm:main-prague}, instead of working directly with the Prague dimension, we use~\eqref{eq:pdim-idim} and upper bound $\mathrm{idim}(G_{n,p})$.
Our main idea is to select independent sets with size proportional to $\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen.
We then control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.

Our strategy is inspired by the work of Bennett and Bohman~\cite{Bennett2016-di}, who use a random greedy independent set process to lower bound the independence number of regular hypergraphs under certain degree and codegree conditions.
For graphs, we also improve on their work by handling pseudorandom graphs with logarithmic degrees rather than regular graphs with degrees growing polynomially with the number of vertices.
Our analysis is done via the differential equation method, which was popularised in the combinatorial community by Wormald~\cite{Wormald-1,Wormald-2}.
As can be seen in the references given in~\cite{Wormald-2}, the roots of the method may be traced back further, see for example the works of Kurtz~\cite{Kurtz1970-aa}
and Karp and Sipser~\cite{Karp1981-aa}.

Our methods also allow us to bound the minimum number of independent sets needed to cover all the non-edges of $G=G_{n,p}$, denoted by $\overline{\theta}_1(G)$.
Initially, one may conjecture that with high probability $\overline{\theta}_1(G)$ should be of order $n^2/\alpha(G)^2$  for all values of $p$, which corresponds to the size of a `near' optimal clique cover of $\overline{G}$.
Frieze and Reed~\cite{Frieze1995-xa}, by improving the results of Bollobás, Erd\H{o}s, Spencer, and West~\cite{Bollobas1993-nl}, showed that this is indeed the case for constant values of $p$.
Surprisingly, this is not true for the sparse range.
In~\cite{Guo2023-wq}, Guo, Patton and Warnke proved that for any fixed $\varepsilon >0$ and $n^{-1} \ll p \le 1-\varepsilon$, with high probability we have
\begin{align*}
	\overline{\theta}_1(G_{n,p}) = \Omega \left ( \dfrac{n^2 \log \frac{1}{p} }{\alpha^2(G_{n,p})} \right ).
\end{align*}
It is worth mentioning that due to the work of Matula~\cite{Matula1970-ka}, Bollobás and Erd\H{o}s~\cite{Bollobas1976-al}, Grimmett and McDiarmid~\cite{Grimmett1975-vs}, and Frieze~\cite{Frieze1990-cu}, we know that the size of the largest independent set in $G_{n,p}$ is of order $\Theta(p^{-1}\log pn)$ for $n^{-1} \ll p \le 1- \varepsilon$, for any fixed $\varepsilon > 0$.
Also note that
$\log \frac{1}{p} = \Theta_{\varepsilon}(\log n)$ for $ n^{-1}\ll p \ll n^{-\varepsilon}$.

Recently, Molnar, Rödl, Sales and Schacht~\cite{MolnarRodlSalesSchacht} obtained a matching upper bound for every $\varepsilon > 0$ and $p$ such that $n^{-1/2} \log^{5/4} n \ll p \ll n^{-\varepsilon}$.
Their argument heavily relies on the fact that the number of independent sets in $G_{n,p}$ is concentrated around its expectation with high probability.
As this no longer holds for $p \ll n^{-1/2}$ (see Corollary~19 in~\cite{Coja-Efthymiou}), there is a barrier at $n^{-1/2}$ for the range of $p$ their method can handle.
Our second main theorem overcomes this natural barrier and extends their result to a much wide range of~$p$.

\begin{theorem}\label{thm:main}
	Let $\varepsilon\in (0,1)$ and $(\log n)^{2+\varepsilon} n^{-1} \le p < n^{-\varepsilon}$.
	Then, with high probability we have
	\begin{align*}
		\overline{\theta}_1(G_{n,p}) = \Theta_{\varepsilon} \left ( \dfrac{n^2\log n}{\alpha^2(G_{n,p})} \right ).
	\end{align*}
\end{theorem}

It is worth noting we were only able to overcome the previous obstacles because for many host graphs the independent set generated by our random greedy process is not uniformly distributed.
As an example, let $1 \ll k \ll n$ and take a complete bipartite graph $K[A,B]$ with $|A| = n/3$ and $|B| = 2n/3$.
Now let $I$ be a uniformly-chosen random independent $k$-set and let $J$ be the independent set generated by our random greedy process.
One can check that if $u, v \in A$, then $\mathbb{P}\left[u,v \in I\right] \ll \mathbb{P}\left[u,v \in J\right]$.

Now let us mention some of the historic background related to $\overline{\theta}_1$.
In the 1960s, Erd\H{o}s, Goodman and Pósa~\cite{Erdos1966-so} introduced the parameter $\theta_1(G)$, defined as the minimum number of cliques required to cover all the edges of $G$.
Notice that $\overline{\theta}_1(G) = \theta_1(\overline{G})$, and hence one may think of $\theta_1(G)$ and $\overline{\theta}_1(G)$ as the dual versions of each other.
In~\cite{Erdos1966-so}, the authors showed that $\theta_1(G) \le \lfloor n^2/4 \rfloor$ for every $n$-vertex graph $G$, which is attained by balanced complete bipartite graphs.
In 1994, Alon~\cite{Alon1986-oc} showed that if $G$ is a $n$-vertex graph with maximum degree $\Delta$, then $\theta_1(\overline{G}) = O(\Delta^2 \log n)$.
Eaton and Rödl~\cite{Eaton1996-ap} showed that this bound is close to best possible, as there are $n$-vertex graphs $G$ with $\theta_1(\overline{G}) = \Omega(\Delta^2 \log n / \log \Delta)$.

Frieze and Reed~\cite{Frieze1995-xa} showed that with high probability $\theta_1(G_{n,p}) = \Theta_p(n^2/\log n)$ for constant values of $p$.
Guo, Patton and Warnke~\cite{Guo2023-wq} extended these results much further by showing that for every fixed $\gamma \in (0,1)$ and $p$ such that $n^{-2} \le p \le 1-\gamma$, with high probability, we have $\theta_1(G_{n,p}) = \Theta (n^2p /\log_{1/p}^{2} n )$.
As results in~\cite{Guo2023-wq} left the range where $p$ tends to 1 open, it is more convenient to just consider $\overline{\theta}_1(G_{n,p})$ for $p$ tending to 0.

The rest of the paper is divided as follows.
In Section \ref{sec:proofs}, we present the proofs of our main theorems; in Section~\ref{sec:random-greedy}, we analyse our random greedy independent set process using the differential equation method; in Section~\ref{sec:prob-cover}, we prove the main technical result of the paper, namely Theorem~\ref{thm:main-dem}; and in Section \ref{sec:preliminaries}, we present some tools and standard lemmas that will be used in the analysis of our random process.

To prove Theorem~\ref{thm:main-prague}, the authors say they do not work directly with the Prague dimension; instead they “use~\eqref{eq:pdim-idim} and upper bound $\mathrm{idim}(G_{n,p})$.” The “main idea” is to “select independent sets with size proportional to $\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen,” and then “control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.” They note the strategy is inspired by Bennett--Bohman’s random greedy independent set process, and that their “analysis is done via the differential equation method.”

To prove the main theorem, the authors say they do not work directly with the Prague dimension; instead they “use
\begin{align}\label{eq:pdim-idim}
	\idim(G) \le \pdim(G) \le \idim(G) + 1.
\end{align}
and upper bound $\mathrm{idim}(G_{n,p})$.” The “main idea” is to “select independent sets with size proportional to $\alpha(G_{n,p})$ via a random greedy process which takes one vertex at a time uniformly at random from the common non-neighbourhood of the vertices already chosen,” and then “control the probability that a given vertex and a given non-edge are covered by an independent set generated by our process.” They note the strategy is inspired by Bennett--Bohman’s random greedy independent set process, and that their “analysis is done via the differential equation method.”

\label{thm:main-prague}
	Let\footnote{In all our statements, $\varepsilon$ is a fixed constant and $n$ tends to infinity.} $\varepsilon \in (0,1)$ and $(\log n)^{2+\varepsilon} n^{-1} \le p < n^{-\varepsilon}$.
	Then, with high probability we have
	\begin{align*}
		\mathrm{pdim}(G_{n,p}) = O_{\varepsilon} \left ( \dfrac{pn \log n}{\log pn} \right ).
	\end{align*},

Question: Let $G_{n,p}$ be the Erd\H{o}s--R\'enyi random graph on $n$ vertices, where each edge is present independently with probability $p$. For a graph $G$, its Prague dimension $\mathrm{pdim}(G)$ is the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph. Fix $\varepsilon \in (0,1)$, assume $n \to \infty$, and suppose the edge probability satisfies $(\log n)^{2+\varepsilon} n^{-1} \le p < n^{-\varepsilon}$. Which quantitative estimate holds with high probability (that is, with probability tending to $1$ as $n\to\infty$)?

With high probability, \[\mathrm{pdim}(G_{n,p}) = O_{\varepsilon}\!\left(\frac{pn\log n}{\log(pn)}\right).\] Equivalently, there exists a constant $C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\frac{pn\log n}{\log(pn)}.\]

Choices:
B. With high probability, \[\mathrm{pdim}(G_{n,p}) = O_{\varepsilon}\!\left(\frac{pn\log n}{\log(1/p)}\right).\] Equivalently, there exists a constant $C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\frac{pn\log n}{\log(1/p)}.\]
C. With high probability, \[\mathrm{pdim}(G_{n,p}) = O_{\varepsilon}(pn\log n).\] Equivalently, there exists a constant $C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\,pn\log n.\]
D. With high probability, \[\mathrm{pdim}(G_{n,p}) = O\!\left(\frac{pn\log n}{\log(pn)}\right).\] Equivalently, there exists an absolute constant $C>0$ independent of $\varepsilon$ such that, with high probability, \[\mathrm{pdim}(G_{n,p}) \le C\frac{pn\log n}{\log(pn)}.\]
E. With high probability, \[\mathrm{pdim}(G_{n,p}) = \Theta_{\varepsilon}\!\left(\frac{pn\log n}{\log(pn)}\right).\] Equivalently, there exist constants $c_{\varepsilon},C_{\varepsilon}>0$ depending only on $\varepsilon$ such that, with high probability, \[c_{\varepsilon}\frac{pn\log n}{\log(pn)}\le \mathrm{pdim}(G_{n,p}) \le C_{\varepsilon}\frac{pn\log n}{\log(pn)}.\]

logarithmic denominator from process length/coverage

dropped the improving factor \(\log(pn)\) in the denominator

dependence of the implied constant on \(\varepsilon\) and the admissible \(p\)-range

upper bound upgraded to matching lower bound without the separate lower-bound argument

The stem does not explicitly reveal the correct option and gives only the hypotheses and terminology needed for the result. There is no direct phrasing that singles out choice A.

This is essentially a theorem-recall item: the stem states the assumptions and asks which asymptotic conclusion holds with high probability. The correct choice is basically the theorem statement itself rather than a conclusion derived from intermediate reasoning.

There is some reasoning pressure because the options differ in subtle but meaningful ways: the denominator, loss of a logarithmic factor, dependence on epsilon, and promotion of an upper bound to a Theta statement. Still, the item mainly tests recognition of the exact result rather than substantial generative reasoning.

The distractors are strong and mathematically plausible. They reflect realistic failure modes: confusing log(pn) with log(1/p), accepting a weaker true bound, overlooking epsilon-dependence of constants, or overstating an upper bound as a matching asymptotic.

A moderately good MCQ with excellent distractors and no answer leakage, but it is largely a direct restatement of a theorem and therefore only weakly tests generative reasoning.


\label{future directions}
In this section, we provide several research questions that allow for potential extensions of this work.

First, we note that our error bounds rely on the assumption that $n

\label{prelim}
Throughout this paper, we will let $q$ be an odd prime power with prime divisor $p$. We will use $\mathcal{M}(n)$ to denote the set of polynomials of degree $n$ in $\mathbb{F}_q[x]$, an

The study of irreducible polynomials with prescribed coefficients over finite fields carries a rich history, with much progress having been made over the last three decades. While the problem of counting irreducible polynomials over finite fields can be traced back to the early 19th century when Gauss proved that there are approximately $q^n/n$ irreducible polynomials of degree $n$ in $\mathbb{F}_q[x]$, this problem with the additional restriction of prescribed coefficients was called into question in 1992 by Hansen and Mullen \cite{Hansen92}, who conjectured that for all prime powers $q \ge 3$ and positive integers $n$, one can find a monic irreducible polynomial of degree $n$ in $\mathbb{F}_q[x]$ with any single coefficient prescribed. In 1997, Wan \cite{Wan97} proved the Hansen-Mullen conjecture for $n \ge 36$ or $q > 19$, leaving finitely many cases remaining; the conjecture was proved in its entirety by Ham and Mullen, who computationally verified the remaining cases.

In general, error bounds on the number of irreducible polynomials with prescribed coefficients have played a key role in the literature of the subject. Their practical usage is mostly present in polynomial existence proving, as proving existence analytically is often much easier compared to direct approaches. However, their ability to represent bias towards some coefficient selections over others is also of interest; in the broader context of arithmetic geometry, for instance, the \textit{analytic rank} is defined to measure the bias of tensors, and has been a topic of interest to many researchers in the field \cite{moshkovitz2024quasilinearrelationpartitionanalytic}.

Since it is hard to understand why singular coefficient prescriptions differ in representation among irreducible polynomials, we are motivated to consider counting irreducible polynomials $a_0+a_1x+\dots+a_{n-1}x^{n-1}+x^n \in \mathbb{F}_q[x]$ whose coefficient vector $(a_0, \dots, a_{n-1}, 1)$ lies in a fixed affine set $V$ of points in $\mathbb{F}_q^{n+1}$, which is the objective of this paper. Indeed, such restrictions have been studied in depth in the context of positive integers and prescribed digits, primarily using tools from analytic number theory. Nevertheless, it is notable that this has also been studied briefly in the finite fields setting, which is often easier to work in than the positive integers; see \cite{Gao2021} for self-reciprocal irreducible polynomials, for example. An important work in this line of research is that of Mérai, who gave error bounds \cite{Mérai2025} for the distribution of the Rudin-Shapiro function over irreducible polynomials in $\mathbb{F}_q[x]$. This was one of the first error bounds obtained on the distribution of a polynomial function in the coefficients of irreducible polynomials over finite fields.

The goal of this paper is as follows.
\begin{quote}
\textsl{Fix polynomials $R_1, \dots, R_m$ on the coefficients of monic polynomials in $\mathbb{F}_q[x]$ of degree $n$. Find error bounds for the number $I_n(R_1, \dots, R_m)$ of monic irreducible polynomials $f$ of degree $n$ such that $(R_1(f), \dots, R_m(f)) = (0, \dots, 0)$.}
\end{quote}
Since on average, the polynomials $R_1$, \dots, $R_m$ take on equidistributed values, we should expect to have \[ I_n(R_1, \dots, R_m) \approx \frac{I(n)}{q^m}, \] where $I(n) = q^n/n + O(q^{n/2}/n)$ is the number of irreducible polynomials of degree $n$ over $\mathbb{F}_q$. Thus, we provide an upper bound on the difference $\left|I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m}\right|$.

\begin{lemma}\label{Vaughan's identity}
 Let $\Psi: \mathbb{F}_q[x] \to \mathbb{C}$ be a function with $|\Psi(f)| \le 1$. If $u$ and $v$ are two integers in $[1, n]$ with $u+v>n$, then \[ \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\Psi(f)\right| \ll n\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\Sigma_2^{1/2}, \] with \[ \Sigma_1 = \sum_{\deg g \le u+v} \left|\sum_{\deg h = n - \deg g} \Psi(gh)\right| \] and \[ \Sigma_2 = \max_{v \le i \le n-u} \max_{\deg g_1 = n-i} \sum_{\deg g_2 = n-i} \left|\sum_{\deg h = i} \Psi(hg_1)\overline{\Psi(hg_2)}\right|, \] where both sums above are taken over monic polynomials.
\end{lemma}
In particular, it suffices to obtain upper bounds on $\Sigma_1$ and $\Sigma_2$. Using polarization, we will reduce the computation to computing character sums on multilinear polynomials.

\begin{lemma}\label{Polarization result}
 Let $P_1(f), \dots, P_m(f)$ be polynomials in the coefficients of a $k$-degree polynomial $f$. Let $j=\max(\deg P_1, \dots, \deg P_m)$ and $S$ denote the set of indices $i \in [1, m]$ for which $\deg P_i = j$. Then we have \[ \left|\sum_{\deg f = k} \prod_{i=1}^m \psi_i(P_i(f))\right| \le \left(q^{k}\left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right|\right)^{1/2^j},\] where the sum above is over monic $f$.
\end{lemma}
\begin{proof}
 Note that \begin{align*}
 \left|\sum_{\deg f = k} \prod_{i=1}^m \psi_i(P_i(f))\right|^2 & = \sum_{\deg f = k} \sum_{\deg h < k} \prod_{i=1}^m \psi_i(P_i(f+h))\overline{\psi_i(P_i(f))} \\ &= \left|\sum_{\deg f = k} \sum_{\deg h < k} \prod_{i=1}^m \psi_i(\Delta_h P_i(f))\right| \\ &\le \sum_{\deg f = k}\left| \sum_{\deg h < k} \prod_{i=1}^m \psi_i(\Delta_h P_i(f))\right|.
 \end{align*}
 Repeatedly squaring in this fashion, we obtain that at the $j$th squaring, \[ \left|\sum_{\deg f = k} \prod_{i=1}^m \psi_i(P_i(f))\right|^{2^j} \le q^{k} \left|\sum_{h_1, \dots, h_j} \prod_{i \in S} \psi_i(P_i^{\circ}(h_1, \dots, h_j)) \right|, \] as desired.
\end{proof}
Recall that the purpose of this application of polarization is to obtain upper bounds on the sums $\Sigma_1$ and $\Sigma_2$ from Lemma \ref{Vaughan's identity} upon selecting $\Psi(f) = \prod_{i=1}^m \psi_i(R_i(f))$. Indeed, we may consider the following choices of $P_i$, which are both polynomials in the coefficients of their arguments.
\begin{itemize}
 \item For $\Sigma_1$, let $(P_i)_g(f)=R_i(fg)$ for all $f \in \mathcal{M}(n-\deg g)$ and $1 \le i \le m$.
 \item For $\Sigma_2$, let $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$ for all $f \in \mathcal{M}(n - \max(\deg g_1, \deg g_2))$ and $1 \le i \le m$.
\end{itemize}

\subsection{Exponential sums of multilinear polynomials}
Note that Lemma \ref{Polarization result} requires us to only work with exponential sums of multilinear polynomials going forward. In order to obtain useful bounds on the exponential sums of multilinear polynomials, we require a corollary of a recent result on the comparison between the analytic rank and partition rank of tensors.
\begin{defn}
 The \emph{analytic rank} $\operatorname{AR}(T)$ of $T$ is defined as \[ \operatorname{AR}(T) := -\log_q\left(\frac{1}{q^n} \sum_{(x_1, \dots, x_n) \in \mathbb{F}_q^n} \psi_0(T(x_1, \dots, x_n))\right). \] The \emph{partition rank} $\operatorname{PR}(T)$ of $T$ is defined as the smallest number $r$ such that $T$ can be written as the sum of $r$ reducible homogeneous polynomials.
\end{defn}

\begin{lemma}\label{char sums and ranks}
For all polynomials $f$ of degree $n<p$, $\operatorname{PR}(f^{\circ}) \ge \operatorname{rank}(f)$. Consequently, for any tuple $(P_1, \dots, P_m)$ of polynomials and nontrivial tuple $(\psi_1, \dots, \psi_m)$ of additive characters on $\mathbb{F}_q$, there exists a constant $c>0$ dependent only on $\max(\deg P_1, \dots, \deg P_m)$ such that \[ \left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right| \le q^{n-c \cdot \operatorname{rank}((P_i)_{i \in S})}, \] where $j = \max(\deg P_1, \dots, \deg P_m)<p$ and $S$ is the set of indices $1 \le i \le m$ for which $\deg P_i = j$.
\end{lemma}
\begin{proof}
Note that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\frac{1}{n!}f^{\circ}(x, \dots x)$. Thus, \[ \operatorname{rank}(f) = \operatorname{rank}(f^{\circ}(x, \dots x)) \le \operatorname{PR}(f^{\circ}), \] proving the first part. Let $L$ denote the set of nontrivial $\mathbb{F}_q$-combinations of $\{P_i: i \in S\}$. Observe that \begin{align*} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right| = \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(\sum_{i \in S} \varphi(\psi_i)P_i^{\circ}(x)\right) \right| \le \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| \end{align*} and \begin{align*} \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| &\le \max_{P \in L} q^{n-\operatorname{AR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{PR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{rank}(P)} \\ &= q^{n-c \cdot \operatorname{rank}((P_i)_{i \in S})}, \end{align*} which proves the second part.
\end{proof}

\section{Proof of main result}\label{main proof}
We are ready to prove our main result.
\begin{proof}[Proof of Theorem \ref{main theorem}]
By the orthogonality of characters, we have \[ I_n(R_1, \dots, R_m) = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m)} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] Thus, \[ I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m} = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] We may rewrite the above as \begin{align*} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)) &= \sum_{f \in \mathcal{M}(n)} \mathbf{1}_{f \in \mathcal{P}(n)}\prod_{i=1}^m \psi_i(R_i(f)) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O\left(\sum_{f \in \mathcal{M}(n)} \mathbf{1}_{\Lambda(f) > 1}\right) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(I(n)-I(n/2)) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(q^{n/2}/n) \end{align*} Therefore, \[ \left|I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m}\right| \le \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^{m} \psi_i(R_i(f))\right| \] by the triangle inequality. By Lemma \ref{Vaughan's identity}, we have for all nontrivial tuples $(\psi_1, \dots, \psi_m)$ of additive characters on $\mathbb{F}_q$ that
\[ \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^m \psi_i(R_i(f))\right| \ll n\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\Sigma_2^{1/2} \]
for all choices of $u$ and $v$ with $u+v<n$, where
\[ \Sigma_1 = \sum_{\deg g \le u+v} \left|\sum_{\deg h = n - \deg g} \prod_{i=1}^m \psi_i((P_i)_g(h)) \right| \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} \left|\sum_{\deg h = k} \prod_{i=1}^m \psi_i((P_i)_{g_1, g_2}(h)) \right|. \]
By Lemma \ref{Polarization result}, we have
\[ \Sigma_1 \le \sum_{0 \le d \le u+v} q^{\frac{n-d}{2^j}} \sum_{\deg g = d} \left|\sum_{x \in (\mathbb{F}_q^d)^{n-d}} \prod_{i \in S} \psi_i((P_i)_g^{\circ}(x))\right|^{1/2^j} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{\frac{k}{2^j}}\left|\sum_{x \in (\mathbb{F}_q^{n-k})^k} \prod_{i \in S} \psi_i((P_i)_{g_1, g_2}^{\circ}(x))\right|^{1/2^j}. \]
Thus, by Lemma \ref{char sums and ranks}, we have
\[ \Sigma_1 = \sum_{0 \le d \le u+v} \sum_{\deg g = d} q^{(n-d)+\frac{n-d}{2^j}-c \cdot \operatorname{rank}(((P_i)_g)_{i \in S})} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{k-c \cdot \operatorname{rank}(((P_i)_{g_1, g_2})_{i \in S})}. \] The desired result follows.
\end{proof}

The structure of this paper is as follows. We begin by stating the preliminaries assumed throughout this paper in Section \ref{prelim}. Next, in Section \ref{lemmas}, we state and prove the lemmas that play a key role in the proof of our main result. Our approach, based on character sums, uses a function field analogue of Vaughan's identity, polarization to reduce the problem to working with multilinear exponential sums, and partition rank to bound the multilinear exponential sums. In Section \ref{main proof}, we prove the main result by applying the lemmas from Section \ref{lemmas}, and we end with Section \ref{future directions}, in which we discuss future research directions.

To prove Theorem~\ref{main theorem}, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from Section~\ref{lemmas}” in Section~\ref{main proof}.

To prove the main theorem, the paper’s approach is “based on character sums” and uses “a function field analogue of Vaughan's identity,” then “polarization to reduce the problem to working with multilinear exponential sums,” and finally “partition rank to bound the multilinear exponential sums.” The main result is then proved “by applying the lemmas from \label{lemmas}
\subsection{Vaughan's identity in $\mathbb{F}_q[x]$}
In \cite{Mérai2025}, the following variant of Vaughan's identity for arithmetic functions on $\mathbb{N}$ is utilized. This bound en” in \begin{proof}
Note that because $n<p$, the degree $n$ homogeneous component of $f$ is given by $\frac{1}{n!}f^{\circ}(x, \dots x)$. Thus, \[ \operatorname{rank}(f) = \operatorname{rank}(f^{\circ}(x, \dots x)) \le \operatorname{PR}(f^{\circ}), \] proving the first part. Let $L$ denote the set of nontrivial $\mathbb{F}_q$-combinations of $\{P_i: i \in S\}$. Observe that \begin{align*} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \prod_{i \in S} \psi_i(P_i^{\circ}(x)) \right| = \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(\sum_{i \in S} \varphi(\psi_i)P_i^{\circ}(x)\right) \right| \le \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| \end{align*} and \begin{align*} \max_{P \in L} \left|\sum_{x \in (\mathbb{F}_q^k)^j} \psi_0\left(P^{\circ}(x)\right) \right| &\le \max_{P \in L} q^{n-\operatorname{AR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{PR}(P^{\circ})} \\ &\le \max_{P \in L} q^{n-c \cdot \operatorname{rank}(P)} \\ &= q^{n-c \cdot \operatorname{rank}((P_i)_{i \in S})}, \end{align*} which proves the second part.
\end{proof}

\section{Proof of main result}\label{main proof}
We are ready to prove our main result.
\begin{proof}[Proof of Theorem \ref{main theorem}]
By the orthogonality of characters, we have \[ I_n(R_1, \dots, R_m) = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m)} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] Thus, \[ I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m} = \frac{1}{q^m} \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)). \] We may rewrite the above as \begin{align*} \sum_{f \in \mathcal{P}(n)} \prod_{i=1}^m \psi_i(R_i(f)) &= \sum_{f \in \mathcal{M}(n)} \mathbf{1}_{f \in \mathcal{P}(n)}\prod_{i=1}^m \psi_i(R_i(f)) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O\left(\sum_{f \in \mathcal{M}(n)} \mathbf{1}_{\Lambda(f) > 1}\right) \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(I(n)-I(n/2))  \\ &= \sum_{f \in \mathcal{M}(n)} \Lambda(f) \prod_{i=1}^m \psi_i(R_i(f)) + O(q^{n/2}/n) \end{align*} Therefore, \[ \left|I_n(R_1, \dots, R_m) - \frac{I(n)}{q^m}\right| \le \sum_{(\psi_1, \dots, \psi_m) \neq \mathbf{0}} \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^{m}  \psi_i(R_i(f))\right| \] by the triangle inequality. By Lemma \ref{Vaughan's identity}, we have for all nontrivial tuples $(\psi_1, \dots, \psi_m)$ of additive characters on $\mathbb{F}_q$ that
\[ \left|\sum_{f \in \mathcal{M}(n)} \Lambda(f)\prod_{i=1}^m \psi_i(R_i(f))\right| \ll n\Sigma_1 + n^{5/2}q^{n-(u+v)/2}\Sigma_2^{1/2} \]
for all choices of $u$ and $v$ with $u+v<n$, where
\[ \Sigma_1 = \sum_{\deg g \le u+v} \left|\sum_{\deg h = n - \deg g} \prod_{i=1}^m \psi_i((P_i)_g(h)) \right| \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} \left|\sum_{\deg h = k} \prod_{i=1}^m \psi_i((P_i)_{g_1, g_2}(h)) \right|. \]
By Lemma \ref{Polarization result}, we have
\[ \Sigma_1 \le \sum_{0 \le d \le u+v} q^{\frac{n-d}{2^j}} \sum_{\deg g = d} \left|\sum_{x \in (\mathbb{F}_q^d)^{n-d}} \prod_{i \in S} \psi_i((P_i)_g^{\circ}(x))\right|^{1/2^j} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{\frac{k}{2^j}}\left|\sum_{x \in (\mathbb{F}_q^{n-k})^k} \prod_{i \in S} \psi_i((P_i)_{g_1, g_2}^{\circ}(x))\right|^{1/2^j}. \]
Thus, by Lemma \ref{char sums and ranks}, we have
\[ \Sigma_1 = \sum_{0 \le d \le u+v} \sum_{\deg g = d} q^{(n-d)+\frac{n-d}{2^j}-c \cdot \operatorname{rank}(((P_i)_g)_{i \in S})} \]
and
\[ \Sigma_2 = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{k-c \cdot \operatorname{rank}(((P_i)_{g_1, g_2})_{i \in S})}. \] The desired result follows.
\end{proof}
”.

\label{main theorem}
  Let $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ odd. Suppose that $n < p$. Let $j:=\max(\deg R_1, \dots, \deg R_m)$, and let $S$ denote the set of indices $1 \le i \le m$ for which $\deg R_i(f) = j$. Define $(P_i)_g(f)=R_i(fg)$ and $(P_i)_{g_1, g_2}(f) = R_i(g_1f) - R_i(g_2f)$. There exists a constant $c>0$, dependent only on $j$, such that \[ \left|I_n(R_1, \dots, R_m)-\frac{I(n)}{q^m}\right| \le \frac{q^m-1}{q^m} \cdot \min_{0 \le u+v < n} \left(nS_1(u, v) + n^{5/2}q^{n-(u+v)/2}\sqrt{S_2(u, v)}\right) \] where \[ S_1(u, v) = \sum_{0 \le d \le u+v} \sum_{\deg g = d} q^{(n-d)+\frac{n-d}{2^j}-c \cdot \operatorname{rank}(((P_i)_g)_{i \in S})} \] and \[ S_2(u, v) = \max_{v \le k \le n-u} \max_{\deg g_1 = n-k} \sum_{\deg g_2 = n-k} q^{k-c \cdot \operatorname{rank}(((P_i)_{g_1, g_2})_{i \in S})}. \],

Existential–Universal
Inequality or Bound

Question:
Let \(\mathbb{F}_q\) be a finite field of odd order \(q\) and characteristic \(p\), and let \(n<p\). Let \(R_1,\dots,R_m\) be polynomial functions of the coefficients of monic degree-\(n\) polynomials over \(\mathbb{F}_q\). Define \(I_n(R_1,\dots,R_m)\) to be the number of monic irreducible polynomials \(f\in \mathbb{F}_q[x]\) of degree \(n\) such that \(R_1(f)=\cdots=R_m(f)=0\), and let \(I(n)\) denote the number of monic irreducible polynomials of degree \(n\) over \(\mathbb{F}_q\). Set \(j:=\max(\deg R_1,\dots,\deg R_m)\), let \(S=\{1\le i\le m: \deg R_i=j\}\), and define \((P_i)_g(f):=R_i(fg)\) and \((P_i)_{g_1,g_2}(f):=R_i(g_1f)-R_i(g_2f)\). For nonnegative integers \(u,v\) with \(u+v<n\), define
\[
S_1(u,v)=\sum_{0\le d\le u+v}\sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\,\operatorname{rank}(((P_i)_g)_{i\in S})}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\max_{\deg g_1=n-k}\sum_{\deg g_2=n-k} q^{k-c\,\operatorname{rank}(((P_i)_{g_1,g_2})_{i\in S})},
\]
where the degree-constrained sums are over monic polynomials. Under these assumptions, which quantitative estimate holds?

There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]

Choices:
B.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v\le n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]
C.
There exists a constant \(c>0\), depending only on \(j\), such that for every pair of nonnegative integers \(u,v\) with \(u+v<n\),
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]
D.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}S_2(u,v)\right).
\]
E.
There exists a constant \(c>0\), depending only on \(j,m\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]

The stem does not explicitly reveal the correct estimate or single out option A. It only sets up notation and asks which uniform bound holds.

This is essentially an exact theorem-recall item: the task is to recognize the correct formal statement of a quoted estimate among near-verbatim variants, rather than derive a new conclusion.

Some careful comparison is needed because the options differ in subtle quantifiers, index sets, and error terms. However, the item mainly tests memory/statement matching, not genuinely generative mathematical reasoning.

The distractors are strong: they are mathematically plausible, close to the target statement, and reflect realistic failure modes such as altered minimization range, replacing a sharp statement by a weaker true one, using the wrong rank family, or losing the square root.

Well-constructed in terms of distractor plausibility and lack of answer leakage, but weak as a reasoning assessment because it is largely a theorem-statement recognition problem.


Question:
Let \(\mathbb{F}_q\) be a finite field of odd order \(q\) and characteristic \(p\), and let \(n<p\). Let \(R_1,\dots,R_m\) be polynomial functions of the coefficients of monic degree-\(n\) polynomials over \(\mathbb{F}_q\). Define \(I_n(R_1,\dots,R_m)\) to be the number of monic irreducible polynomials \(f\in \mathbb{F}_q[x]\) of degree \(n\) such that \(R_1(f)=\cdots=R_m(f)=0\), and let \(I(n)\) denote the number of monic irreducible polynomials of degree \(n\) over \(\mathbb{F}_q\). Set \(j:=\max(\deg R_1,\dots,\deg R_m)\), let \(S=\{1\le i\le m: \deg R_i=j\}\), and define \((P_i)_g(f):=R_i(fg)\) and \((P_i)_{g_1,g_2}(f):=R_i(g_1f)-R_i(g_2f)\). For nonnegative integers \(u,v\) with \(u+v<n\), define
\[
S_1(u,v)=\sum_{0\le d\le u+v}\sum_{\deg g=d} q^{(n-d)+\frac{n-d}{2^j}-c\,\operatorname{rank}(((P_i)_g)_{i\in S})}
\]
and
\[
S_2(u,v)=\max_{v\le k\le n-u}\max_{\deg g_1=n-k}\sum_{\deg g_2=n-k} q^{k-c\,\operatorname{rank}(((P_i)_{g_1,g_2})_{i\in S})},
\]
where the degree-constrained sums are over monic polynomials. Under these assumptions, which quantitative estimate holds?

Choices:
B.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v\le n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]
C.
There exists a constant \(c>0\), depending only on \(j\), such that for every pair of nonnegative integers \(u,v\) with \(u+v<n\),
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]
D.
There exists a constant \(c>0\), depending only on \(j\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \frac{q^m-1}{q^m}\cdot \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}S_2(u,v)\right).
\]
E.
There exists a constant \(c>0\), depending only on \(j,m\), such that
\[
\left|I_n(R_1,\dots,R_m)-\frac{I(n)}{q^m}\right|\le \min_{0\le u+v<n}\left(nS_1(u,v)+n^{5/2}q^{\,n-(u+v)/2}\sqrt{S_2(u,v)}\right).
\]

The stem does not reveal the correct option directly; it only sets up notation and hypotheses. The answer must be identified from the choices.

This is essentially a theorem-statement recall item: after presenting all assumptions and definitions, it asks which quantitative estimate holds. The correct choice is basically the theorem’s conclusion verbatim.

Some care is needed to distinguish subtle variants involving the minimization, the range of u+v, the square root on S_2, and constant dependence. However, the task is still mostly recognition/recall of the exact statement rather than genuine derivation or synthesis.

The distractors are mathematically plausible and target realistic failure modes: changing the admissible range, dropping the minimization, removing the square root, or altering the constant/prefactor. They are distinct and nontrivial.

A strong theorem-precision MCQ with high-quality distractors and no answer leakage, but it is largely a direct restatement/recognition task rather than a genuinely generative reasoning problem.


\label{s:data}

\Cref{table:triples} contains \emph{all} triples of pairs of integers, $(p_1,q_1)$, $(p_2,q_2)$, $(p_3,q_3)$ such that:
\begin{itemize}
\item there exist rational homology balls $B_1$,

\begin{proof}[Proof of Corollaries \ref{cor:Bpq} and \ref{cor:Lpq}]
For this we note that by \Cref{thm:2Farey}, either $\dfrac{p}{q}$ or $\dfrac{p}{p-q}$ appears in a 2-Farey triple.  It follows that $B_{p,q}$ embeds smoothly in $\CPb$ and hence so does its boundary $L(p^2,pq-1)$.  Finally note that an embedding of a 3-manifold into a 4-manifold does not depend on the orientation of either.
\end{proof}

\section{A common framework for known results}\label{s:compare}

In this section we make an observation which attempts to put \Cref{thm:ANN} in a common framework with previous results of Hacking--Prokhorov \cite{hp}, Evans--Smith \cite{es}, and Lisca--Parma \cite{HD2}.  Each of these results gives a binary tree, together with an embedding of a triple of rational balls in $\CP$ for each node of the tree.  The first example of this is the famous Markov tree which contains all triples of solutions of
\begin{equation*}
p_1^2+p_2^2+p_3^2=3p_1p_2p_3.
\end{equation*}
This features in each of \cite{es,hp,HD2}.  Two further trees of triples arise from \cite{HD2}, and the 2-Farey tree underlies \Cref{thm:ANN}.  The following definition may be interpreted in terms of the proofs of \Cref{prop:2Femb} and  \cite[Proposition 4.6]{HD1}.  In each case we have in mind three curves lying on or near nested tori, and the recursive rule is given by handlesliding one curve over another.

\begin{defn}
A \emph{signed slide triple tree} is an infinite rooted binary tree with a triple $\left(\left(\dfrac{p_1}{q_1},\delta_1\right),\left(\dfrac{p_2}{q_2},\delta_2\right),\left(\dfrac{p_3}{q_3},\delta_3\right)\right)$ at each node and the recursive rule

{\centering
\includegraphics[width=\textwidth]{./figures/sliderule.pdf}
\par}
\noindent
where $\delta_i\in\{\pm1\}$, $x_1=p_2q_3-p_3q_2$ and $x_3=p_1q_2-p_2q_1$.
\end{defn}

We note that the following gives new recursive descriptions of the triples appearing in \cite[Theorem 1.2(2),(3)]{HD2}.
\begin{thm}\label{thm:trees}
Let $\left(\left(\dfrac{p_1}{q_1},\delta_1\right),\left(\dfrac{p_2}{q_2},\delta_2\right),\left(\dfrac{p_3}{q_3},\delta_3\right)\right)$ be a triple appearing at a node of a signed slide triple tree whose root is labeled by one of
\begin{enumerate}
\item \triple{1}{-1}{1}{5}{1}{1}{2}{1}{1},
\item \triple{1}{1}{-1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{-1}{1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{0}{-1}{3}{2}{-1}{2}{2}{-1}.
\end{enumerate}
Then the disjoint union $\bigsqcup\delta_i B_{p_i,q_i}$ embeds smoothly in $\CP$.  The first root above gives the triples arising from the Markov tree, the second and third are the second and third families of embeddings in \cite[Theorem 1.2]{HD2}, and the fourth gives the triples in \Cref{thm:ANN}.
\end{thm}

\begin{proof}[Proof sketch]
For the case of the first root, which yields the Markov tree, we appeal to \cite{perling}, which refers to $\{q_1,q_2,q_3\}$ as the \emph{T-weights} of the Markov triple $\{p_1,p_2,p_3\}$.  The identities \cite[(1)]{perling} show that $x_1=3p_1$ and $x_3=3p_3$, and then the recursive rule above agrees with the mutation rules for Markov triples and their T-weights \cite[\S2 and Lemma 3.1]{perling}.  The first three rows of the tree are shown below.  We omit the $\delta_i$ which are all equal to $+1$.

{\centering
\includegraphics[width=10cm]{./figures/MarkovTree.pdf}
\par}
\noindent

The second root above yields the triples of rational balls embedding in $\CP$ given by Lisca--Parma in \cite[Theorem 1.2(2)]{HD2}.
One can show that $x_1=\delta_1p_1$ and $x_3=\delta_3p_3$ and that, as for the standard Markov equation, the recursive rule given above agrees with the mutation rule for the triples of solutions and corresponding weights of the Markov-type equation
\begin{equation*}
\delta_1 p_1^2+\delta_2 p_2^2 +\delta_3 p_3^2=p_1 p_2 p_3,\end{equation*}
with $\{\delta_1, \delta_2, \delta_3\}=\{1,-1,-1\}$ and weights
\begin{align*}
q_1&\equiv\delta_2p_2/p_3\equiv-\delta_3p_3/p_2\pmod{p_1},\\
q_2&\equiv\delta_3p_3/p_1\equiv-\delta_1p_1/p_3\pmod{p_2},\\
q_3&\equiv\delta_1p_1/p_2\equiv-\delta_2p_2/p_1\pmod{p_3}.
\end{align*}
Each weight $q_i$ is strictly between $0$ and $p_i$ unless $p_i=1$, in which case we take $q_i=1$.

The first entries of the resulting tree of triples are shown below.  In each triple the underlined entry has $\delta_i=1$.  Note that $B_{-p,-q}=B_{p,q}$, but the negative signs in the tree below are needed for the recursion.

{\centering
\includegraphics[width=10cm]{./figures/HD2secondtree.pdf}
\par}
\noindent

The first few entries of the tree arising from the third root in the statement are shown below, as well as two extra nodes before the root.  In each triple the underlined entry has $\delta_i=-1$.

{\centering
\includegraphics[width=10cm]{./figures/HD2thirdtree.pdf}
\par}
\noindent
We claim that this contains precisely the triples of rational balls embedding in $\CP$ given by Lisca--Parma in \cite[Theorem 1.2(3)]{HD2}.  To see this, for each node, let $\gamma_i=(p_i,q_i)$,  and take
$$(x_1,x_2,x_3)=(\gamma_2\cdot\gamma_3,\gamma_1\cdot\gamma_3,\gamma_1\cdot\gamma_2),$$
where $(p,q)\cdot(r,s)=ps-qr$.  For example, if we consider the first triple
$\left(\frac{1}{-1},\frac{1}{\underline{0}},\frac{-1}{-1}\right)$, we get
$$\gamma_1=(1,-1), \gamma_2=(1,0), \gamma_3=(-1,-1) \text{ and } (x_1,x_2,x_3)=(-1,-2,1),$$
and for the second triple $\left(\frac{1}{-1},\frac{2}{1},\frac{1}{\underline{0}}\right)$ we obtain
$$\gamma_1=(1,-1), \gamma_2=(2,1), \gamma_3=(1,0) \text{ and } (x_1,x_2,x_3)=(-1,-1,-3).$$

We claim that every triple in this tree satisfies the following Markov-type equation
\begin{equation}
\delta_1 x_1^2+\delta_2 x_2^2 +\delta_3 x_3^2=x_1 x_2 x_3-4,\label{eq:3-2}
\end{equation}
and also
\begin{equation}
\delta_1 p_1^2+\delta_2 p_2^2 +\delta_3 p_3^2-\delta_1\delta_2p_1 p_2 x_3-\delta_1\delta_3p_1p_3x_2-\delta_2\delta_3p_2p_3x_1-p_1p_3x_1x_3=0.\label{eq:3-3}
\end{equation}
These are equations (3-2) and (3-3) in \cite{HD2}.

It is straightforward to check these equations are satisfied for the triple $\left(\frac{1}{-1},\frac{1}{\underline{0}},\frac{-1}{-1}\right)$, which forms the base case for a structural induction.  We then check that triples given by the recursive rule continue to satisfy \eqref{eq:3-2} and \eqref{eq:3-3}.

Suppose that we have $\delta_2=-1$ and that
$$\gamma_1=(p_1,q_1), \gamma_2=(p_2,q_2), \gamma_3=(p_3,q_3),$$
and thus
$$(x_1,x_2,x_3)=(p_2q_3-p_3q_2,p_1q_3-p_3q_1,p_1q_2-p_2q_1).$$
By induction, we assume that \eqref{eq:3-2} and \eqref{eq:3-3} hold for these variables.
Moving left down the tree, the recursive rule leads to $\delta_3=-1$,
$$\widehat{\gamma}_1=(p_1,q_1), \widehat{\gamma}_2=(-p_3-x_1p_2,-q_3-x_1q_2), \widehat{\gamma}_3=(p_2,q_2),$$
and
\begin{equation}
(\widehat{x}_1,\widehat{x}_2,\widehat{x}_3)=(x_1,x_3,-x_2-x_1x_3).\label{eq:mut}
\end{equation}
We verify that \eqref{eq:3-2} holds:
\begin{align*}
&\widehat{x}_1^2+\widehat{x}_2^2-\widehat{x}_3^2-\widehat{x}_1\widehat{x}_2\widehat{x}_3+4\\
&=x_1^2+x_3^2-x_2^2-2x_1x_2x_3-x_1^2x_3^2+x_1x_2x_3+x_1^2x_3^2+4\\
&=x_1^2-x_2^2+x_3^2-x_1x_2x_3+4\\
&=0,
\end{align*}
and also \eqref{eq:3-3}:
\begin{align*}
&\widehat{p}_1^2+\widehat{p}_2^2-\widehat{p}_3^2-\widehat{p}_1 \widehat{p}_2 \widehat{x}_3+\widehat{p}_1\widehat{p}_3\widehat{x}_2+\widehat{p}_2\widehat{p}_3\widehat{x}_1-\widehat{p}_1\widehat{p}_3\widehat{x}_1\widehat{x}_3\\
&=p_1^2+p_3^2+2p_2p_3x_1+p_2^2x_1^2-p_2^2-p_1p_3x_2-p_1p_2x_1x_2-p_1p_3x_1x_3-p_1p_2x_1^2x_3\\
&\quad+p_1p_2x_3-p_2p_3x_1-p_2^2x_1^2+p_1p_2x_1x_2+p_1p_2x_1^2x_3\\
&=p_1^2-p_2^2+p_3^2+p_1p_2x_3-p_1p_3x_2+p_2p_3x_1-p_1p_3x_1x_3\\
&=0,
\end{align*}
in both cases using induction for the final equality.

The other cases (moving right down the tree from $\delta_2=-1$, or starting with $\delta_2=1$) are minor variants of this.

The proof of \cite[Theorem 1.2(3)]{HD2} shows that the triples therein come from solutions to \eqref{eq:3-2} and \eqref{eq:3-3} such that the triples $(x_1,x_2,x_3)$ are obtained from the minimal solution $(1,2,1)$ by mutations, sign changes, and permutations as in \eqref{eq:mut}.  By comparison, we conclude that our  signed slide triple tree contains all of the triples of rational balls in the family given in \cite[Theorem 1.2(3)]{HD2}.

For the last case, one can show by induction that $x_1=x_3=2$ for all nodes, and then the recursion rule is easily seen to agree with that of the 2-Farey tree.
\end{proof}

\label{s:balls}
We use Hirzebruch--Jung continued fractions, with the following notation:
$$[a_1,a_2,\ldots,a_k]
:=a_1-\frac{1}{a_2-\raisebox{-3mm}{$\ddots$
\raisebox{-2mm}{${-\frac{1}{\displaystyle{a

\label{s:homotopyCP2s}

In this section we describe some methods for finding rational balls bounded by lens spaces which embed in a homotopy $\CP$ or $\CPb$.
These methods are suitable for use in a co

\begin{quest}\label{q:exoticCP2s}
Are all homotopy $\CP$'s constructed in this section diffeomorphic to the standard $\CP$?
\end{quest}

\begin{proof}
First note that by reversing the first diagram in either \Cref{fig:Bpq} or \ref{fig:Apq}, we see that $B_{p,q}\cong B_{p,p-q}$.

Now take $k\in\Z$.  Start with $B_{p,kp+q}$ as in the second diagram of either \Cref{fig:Bpq} or \ref{fig:Apq}, with the dotted circle replaced by a $\langle0\rangle$-framed circle as in the proof of Lemma $\ref{lem:Bpq}$.  Then the  diffeomorphism $B_{p,q}\cong B_{p,kp+ q}$ follows by performing Rolfsen twists \cite[Section 5.3]{GS} along this $\langle0\rangle$-framed circle.
\end{proof}

\section{The 2-Farey tree}\label{s:2Fareytree}

We define a variant of the classical Farey tree \cite{aigner}, which we refer to as the 2-Farey tree.  This is a binary tree with ordered triples of fractions, called 2-Farey triples, at each node.  The triple at the root is $\left(\frac10,\frac32,\frac22\right)$ and the recursive rule is given by

{\centering
\includegraphics[width=7cm]{./figures/2Fareyrule.pdf}
\par}
\noindent

The first three rows of the tree, and two further nodes, are shown below.

{\centering
\includegraphics[width=10cm]{./figures/2Fareytree.pdf}
\par}
\noindent
Observe that each triple contains two reduced fractions and one fraction $p/q$ with $\gcd(p,q)=2$.

For comparison, part of the standard Farey tree is shown below.  It uses the same recursive rule as above but the triple at the root is $\left(\frac10,\frac11,\frac01\right)$.

{\centering
\includegraphics[width=10cm]{./figures/Fareytree.pdf}
\par}
\noindent
Note that the 2-Farey tree is obtained from the subtree of the Farey tree on the left, beginning at $\left(\frac10,\frac31,\frac21\right)$, by multiplying all denominators by 2.

The following theorem  follows immediately from the corresponding facts for the classical Farey tree found in the wonderful book by Aigner \cite[Section 3.2]{aigner}.

\begin{thm}\label{thm:2Farey}
Every fraction $p/q$ with $q$ even, $p\ge q$, and $\gcd(p,q)\le2$ appears in the $2$-Farey tree.  Every such fraction with $p>2$ is generated as a mediant exactly once.  For each $2$-Farey triple $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dfrac{p_3}{q_3}\right)$, we have
\begin{equation}\label{eqn:2Farey}
p_2=p_1+p_3,\quad q_2=q_1+q_3,\quad \mathrm{and}\quad p_iq_{i+1}-p_{i+1}q_i=2\ \mathrm{for}\  i\in\{1,2\}.
\end{equation}
Furthermore:

\begin{enumerate}[(i)]
\item If $p_1>q_1$ and $p_2>q_2$ satisfy $q_i$ even and $p_1q_2-p_2q_1=\pm2$, then $p_1/q_1$ and $p_2/q_2$ are contained in a $2$-Farey triple.
\item For any coprime natural numbers $p_1, p_2$, there exist unique even numbers $q_1$ and $q_2$ satisfying
$0\le q_i\le p_i$ and $p_1q_2-p_2q_1=\pm2$.  After possibly relabeling, we may take $p_1q_2-p_2q_1=2$, and then $\left(\dfrac{p_1}{q_1},\dfrac{p_1+p_2}{q_1+q_2},\dfrac{p_2}{q_2}\right)$ is a $2$-Farey triple.
\end{enumerate}
\end{thm}

\begin{rmk}
One could alternatively choose the triple $\left(\frac20,\frac31,\frac11\right)$ at the root of the 2-Farey tree.  The resulting tree would be obtained from that above by replacing each fraction $p/q$ by $p/(p-q)$, and reflecting the tree and each triple from left to right.  It contains all triples $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dfrac{p_3}{q_3}\right)$ with $p_i\equiv q_i \pmod2$ and  satisfying \eqref{eqn:2Farey}.
Since by \Cref{lem:addp} we have $B_{p,q}=B_{p,p-q}$, this would result in the same embeddings of rational balls in $\CPb$ in the next section.

We could also consider an expanded 2-Farey tree obtained by multiplying denominators by 2 in all nodes of the full Farey tree shown above.  It again follows from \Cref{lem:addp} that this does not yield any more embedding results than those coming from the 2-Farey tree we have selected, though we will see that for the base of our induction, it is helpful to include two more nodes at the base of the tree.
\end{rmk}

\section{Embeddings of 2-Farey triples}\label{s:ANN}

Given an ordered $n$-tuple $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dots,\dfrac{p_n}{q_n}\right)$, with $\gcd(p_i,q_i)\in\{1,2\}$, we define a Kirby diagram of a manifold $X=X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$ as follows.  Let $r_i=1-i/(n+1)$ for $i=1,\dots,n$, so that
$$1>r_1>r_2>\dots>r_n>0,$$
and $\varepsilon=1/(4n)$, so that the intervals $[r_i-\varepsilon,r_i+\varepsilon]$ are pairwise disjoint in $(0,1)$.

The manifold $X$ is a handlebody with one $0$-handle, one $1$-handle, and $n$ $2$-handles.  To draw the diagram, start with a dotted circle drawn as a circle of diameter 3.  For each $1\le i\le n$, draw the framed knot $K_{p_i,q_i}$ from \Cref{s:balls} in an $\varepsilon$-neighbourhood of the torus which is distance $r_i$ from the dotted circle.  An example is shown in Figure \ref{fig:Base}.

\begin{figure}[htbp]
\centering
\includegraphics[width=14cm]{./figures/Base.pdf}
\caption{{\bf The manifold $X_{\frac10,\frac12,\frac02}$.} The black arrow indicates a handle slide that may be used to simplify the diagram.}
\label{fig:Base}
\end{figure}

\begin{lemma}\label{lem:LPembed}
For any $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dots,\dfrac{p_n}{q_n}\right)$, with $\gcd(p_i,q_i)\in\{1,2\}$, the disjoint union
$$\bigsqcup_{i=1}^n B_{p_i,q_i}$$
embeds smoothly in $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$.
\end{lemma}
\begin{proof}
This follows exactly as in the proof of \cite[Lemma 4.1]{HD2}.  We give a sketch here for convenience.  Consider the 4-manifold given by a single dotted circle, as found in each of $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$ and $B_{p_i,q_i}$.  This is a copy of $S^1\times B^3$.  For convenience, view $B^3$ as the unit ball in $\R^3$.  Then a torus of constant distance $r$ from the dotted circle corresponds to $S^1\times S_\zeta$, where $S_\zeta$ is the intersection of the unit sphere with the plane $z=\zeta$.  Here we  take $\zeta(r)$ to be a monotone decreasing bijection from $(0,\infty)$ to $(-1,1)$.

For each $1\le i\le n$, the set of points $(x,y,z)\in B^3$ with $z\in [\zeta(r_i-\varepsilon),\zeta(r_i+\varepsilon)]$ is again a 3-ball, after smoothing corners, which we may denote by $B^3_i$.  The disjoint union $\bigsqcup_{i=1}^n S^1\times B^3_i$ is embedded in $S^1\times B^3$, and since the 2-handle for each $B_{p_i,q_i}$ is attached to the boundary of $S^1\times B^3_i$, the statement follows.
\end{proof}

\label{s:intro}
A basic, yet fundamental, question in 4-manifold topology is the following.
Given a smooth 4-manifold $X$ and a class $\mathcal{Y}$ of 3-manifolds, which $Y \in \mathcal{Y}$ smoothly embed in $X$?
In this paper, we focus on the case when $\mathcal{Y}$ is the class of lens spaces, which are the simplest 3-manifolds after the sphere, and on the simplest possible 4-manifold in which they can embed\footnote{It is known since Hantzsche's work~\cite{Hantzsche} that lens spaces cannot embed in $S^4$.}: the complex projective plane $\CP$.

Whenever a lens space $L$ embeds in $\CP$, it splits it into two components, one of which is a \emph{rational homology ball}, a 4-manifold with the rational homology of a point.
Therefore, asking for the embedding of $L$ into $\CP$ is equivalent to asking for the embedding into $\CP$ of a rational ball whose boundary is $L$.
Note that lens spaces that bound rational homology balls have been classified by Lisca~\cite{Lisca-ribbon}.

Early results of the existence of this type of embedding come from algebraic geometry~\cite{Manetti, hp} and symplectic topology~\cite{es}, as well as from~\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.
These results indicate that it is interesting to look at embeddings of \emph{triples} of rational homology balls in $\CP$, a question which is related (but not equivalent) to asking about disjoint embeddings of lens spaces.

Here we push the ideas of~\cite{HD2} and of~\cite{EMPR} further, and we obtain many more embeddings of \emph{triples} of rational homology balls with lens space boundary in $\CPb$;
reversing all orientations gives embeddings in $\CP$.
Our triples give previously unknown examples, except possibly when $\min\{p_i\} \le 2$.

Early results of the existence of this type of embedding come from algebraic geometry~\cite{Manetti, hp} and symplectic topology~\cite{es}, as well as from~\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.
These results indicate that it is interesting to look at embeddings of \emph{triples} of rational homology balls in $\CP$, a question which is related (but not equivalent) to asking about disjoint embeddings of lens spaces.

When $p$ and $q$ are coprime or have greatest common divisor 2, $B_{p,q}$ is a rational homology ball constructed with only one 1-handle and one 2-handle, whose boundary is $L(p^2,pq-1)$.
Kirby diagrams for these rational balls are given in Figures~\ref{fig:Bpq} and~\ref{fig:Apq}.

\begin{maincor}\label{cor:Bpq}
Let $p\ge q$ be nonnegative integers. If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the rational ball $B_{p,q}$ admits a smooth orientation-preserving embedding in $\CPb$.
\end{maincor}

\begin{maincor}\label{cor:Lpq}
Let $p\ge q$ be nonnegative integers. If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the lens space $L(p^2,pq-1)$ admits a smooth embedding in $\CP$.
\end{maincor}

\begin{enumerate}[(i)]
\item If $p_1>q_1$ and $p_2>q_2$ satisfy $q_i$ even and $p_1q_2-p_2q_1=\pm2$, then $p_1/q_1$ and $p_2/q_2$ are contained in a $2$-Farey triple.
\item For any coprime natural numbers $p_1, p_2$, there exist unique even numbers $q_1$ and $q_2$ satisfying
$0\le q_i\le p_i$ and $p_1q_2-p_2q_1=\pm2$. After possibly relabeling, we may take $p_1q_2-p_2q_1=2$, and then $\left(\dfrac{p_1}{q_1},\dfrac{p_1+p_2}{q_1+q_2},\dfrac{p_2}{q_2}\right)$ is a $2$-Farey triple.
\end{enumerate}
\end{thm}

\begin{lemma}\label{lem:LPembed}
For any $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dots,\dfrac{p_n}{q_n}\right)$, with $\gcd(p_i,q_i)\in\{1,2\}$, the disjoint union
$$\bigsqcup_{i=1}^n B_{p_i,q_i}$$
embeds smoothly in $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$.
\end{lemma}
\begin{proof}
This follows exactly as in the proof of \cite[Lemma 4.1]{HD2}. We give a sketch here for convenience. Consider the 4-manifold given by a single dotted circle, as found in each of $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$ and $B_{p_i,q_i}$. This is a copy of $S^1\times B^3$. For convenience, view $B^3$ as the unit ball in $\R^3$. Then a torus of constant distance $r$ from the dotted circle corresponds to $S^1\times S_\zeta$, where $S_\zeta$ is the intersection of the unit sphere with the plane $z=\zeta$. Here we take $\zeta(r)$ to be a monotone decreasing bijection from $(0,\infty)$ to $(-1,1)$.

We note that the following gives new recursive descriptions of the triples appearing in \cite[Theorem 1.2(2),(3)]{HD2}.
\begin{thm}\label{thm:trees}
Let $\left(\left(\dfrac{p_1}{q_1},\delta_1\right),\left(\dfrac{p_2}{q_2},\delta_2\right),\left(\dfrac{p_3}{q_3},\delta_3\right)\right)$ be a triple appearing at a node of a signed slide triple tree whose root is labeled by one of
\begin{enumerate}
\item \triple{1}{-1}{1}{5}{1}{1}{2}{1}{1},
\item \triple{1}{1}{-1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{-1}{1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{0}{-1}{3}{2}{-1}{2}{2}{-1}.
\end{enumerate}
Then the disjoint union $\bigsqcup\delta_i B_{p_i,q_i}$ embeds smoothly in $\CP$. The first root above gives the triples arising from the Markov tree, the second and third are the second and third families of embeddings in \cite[Theorem 1.2]{HD2}, and the fourth gives the triples in \Cref{thm:ANN}.
\end{thm}

\begin{prop}\label{prop:Addc}
Suppose that we have two rational numbers, $p/q = [a_1,\dots,a_m]$ and $r/s = [b_1,\dots,b_n]$, and an integer $c$ such that:
\begin{itemize}
\item $L(p,q)$ and $L(r,s)$ bound rational homology balls,
\item letting $t/u = [a_m,\dots,a_1,c,b_1,\dots,b_n]$, $L(t,u)$ also bounds a rational homology ball, and
\item $p$, $r$, and $t$ are pairwise coprime.
\end{itemize}
Then there exists a homotopy $\CP$ or $\CPb$, $X$, and an embedding $B \sqcup B' \sqcup (-B'') \hookrightarrow X$, where $B$, $B'$, and $B''$ are three rational homology balls bounding $L(p,q)$, $L(r,s)$, and $L(t,u)$, respectively, constructed using handles of index at most $2$.

\begin{figure}[htbp]
\centering
\includegraphics[width=14cm]{./figures/Apq.pdf}
\caption{{\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\gcd(p,q)=2$.} The blue attaching circle for the 2-handle in the second diagram is the $(2,-1)$ cable, relative to the torus framing, of the torus knot $T_{-p/2,q/2}$.}
\label{fig:Apq}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=14cm]{./figures/Bpq.pdf}
\caption{{\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\gcd(p,q)=1$.} The second diagram should be closed like a braid closure, so that there is one dotted circle in black and one 2-handle attaching circle, which is a copy of the torus knot $T_{-p,q}$, in blue.  The framing on the 2-handle is $-1$ relative to the torus framing.}
\label{fig:Bpq}
\end{figure}

\begin{mainthm}\label{thm:ANN}
If $p_1,p_2,q_1,q_2$ are nonnegative integers with $q_1, q_2$ even, $p_i\ge q_i$, and $p_1q_2-p_2q_1=\pm2$, then the disjoint union
\[
B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,q_1+q_2}
\]
admits a smooth orientation-preserving embedding in $\CPb$.
\end{mainthm}

When $p$ and $q$ are coprime or have greatest common divisor 2, $B_{p,q}$ is a rational homology ball constructed with only one 1-handle and one 2-handle, whose boundary is $L(p^2,pq-1)$.
Kirby diagrams for these rational balls are given in Figures~\ref{fig:Bpq} and~\ref{fig:Apq}.

An immediate corollary of Theorem~\ref{thm:ANN} above is the following.

\begin{maincor}\label{cor:Bpq}
Let $p\ge q$ be nonnegative integers.  If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the rational ball $B_{p,q}$ admits a smooth orientation-preserving embedding in $\CPb$.
\end{maincor}

In terms of embeddings of lens spaces, the following corollary shows that, among lens spaces of the form $L(p^2,pq-1)$ that bound a rational homology ball,
roughly three-quarters  embed in $\CP$.

\begin{maincor}\label{cor:Lpq}
Let $p\ge q$ be nonnegative integers.  If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the lens space $L(p^2,pq-1)$ admits a smooth embedding in $\CP$.
\end{maincor}

The inspiration for the main theorem comes from the data extracted from the output of two more general constructions, that we describe in detail in Section~\ref{s:homotopyCP2s}, which give a large collection of embeddings of triples of rational homology balls in  homotopy $\CP$'s.
We expect that all these homotopy $\CP$'s are in fact standard (see \Cref{q:exoticCP2s} below).

In a companion paper~\cite{GOobst},  we provide obstructions to the existence of embeddings of such triples, and work in the direction of  a conjecture of Koll\'ar \cite{Kollar}; see also~\cite{JPP} for related recent work. It is worth noting that the embeddings of triples of Theorem~\ref{thm:ANN} account for a very large fraction of the unobstructed triples.

The proof of Theorem~\ref{thm:ANN} is inductive, and relies on a tricky, but rather direct, Kirby calculus argument.
It uses structural induction on a simple variant of the Farey tree, which we will define in Section~\ref{s:2Fareytree}.
The Kirby calculus argument is related to Lisca and Parma's horizontal decompositions, and is informed by the Kirby diagrams of~\cite{EMPR}.
The main idea is to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\times D^3$.

The handlesliding used to prove Theorem~\ref{thm:ANN} suggests another point of view on the results of Lisca and Parma in \cite{HD2}.  In \Cref{thm:trees} we use this point of view to derive new recursive descriptions of the triples of rational balls embedded in $\CP$ obtained in that paper.

\subsection*{History and motivation}
As mentioned above, the motivation for studying embeddings of lens spaces in $\CP$ comes from algebraic geometry.
Inspired by consequences of the orbifold Bogomolov--Miyaoka--Yau inequality, Koll\'ar asks in~\cite{Kollar} whether every compact, orientable, simply-connected 4-manifold $M$ with $H_1(M) = 0$, $H_2(M)\cong\mathbb{Z}$, and with boundary a union of spherical 3-manifolds $Y_1, \dots, Y_n$, satisfies:
\[
\sum_{i=1}^n \left(1-\frac1{|\pi_1(Y_i)|}\right) \le 3.
\]
Lens spaces are finite cyclic quotients of $S^3$, so they are spherical 3-manifolds, and indeed the complements of the rational homology balls that we embed in $\CP$ are manifolds satisfying the assumption in Koll\'ar's question.

A prominent family of examples of such manifolds originates from weighted projective planes $\mathbb{P}(a \sd b \sd c)$ by removing small open neighbourhoods of their (potentially) singular points $(0\sd 0\sd 1)$, $(0\sd 1\sd 0)$, and $(1\sd 0\sd 0)$.
When $(a,b,c)$ satisfies $a^2 + b^2 + c^2 = 3abc$, the weighted projective plane $\mathbb{P}(a^2 \sd b^2 \sd c^2)$ admits a deformation to $\CP$.
Such triples are called \emph{Markov triples}, and the deformation gives rise to an embedding of three rational homology balls, each with boundary a lens space, into $\CP$.
Whenever an embedding of this type exists, the orders of the fundamental groups are coprime perfect squares, and the inequality conjectured by Koll\'ar implies that $M$ has at most three boundary components.

In the realm of complex algebraic geometry, this is in fact the only instance of such embeddings, as was shown by Hacking and Prokhorov~\cite{hp}.
The corresponding result in the symplectic category is also known to hold, by work of Evans and Smith~\cite{es}.
Explicit Kirby diagrams for the associated 4-manifolds with boundary were exhibited by Etnyre, Min, Piccirillo, and Roy~\cite{EMPR}.
Topologically, however, the situation is quite different: earlier work of the second author~\cite{nonsymp},
 and of Lisca and Parma~\cite{LPStein,HD1, HD2}, shows that there are infinitely many embeddings of rational balls in $\CP$ that do not arise symplectically.

We would like to stress that, among the rational homology balls $\pm B_{p,q}$, the only ones that carry some `good' complex or symplectic structure are the those for which $\gcd(p,q) = 1$, with the positive sign.
By `good' here we mean that either they are diffeomorphic to a Milnor fibre in the complex setup \cite{wahl}, or that they have a symplectic structure with respect to which the boundary is convex in the symplectic setup \cite{LiscaSympFill}.
In particular, since in Theorem~\ref{thm:ANN} we have embeddings of $B_{p,q}$ in $\CPb$, none of the nontrival rational homology balls in any of the triples, with the exception of $B_{2,0}$, bear any complex or symplectic significance.

\subsection*{Notation and conventions}
Unless otherwise stated, homology is taken with integer coefficients. The lens space $L(p,q)$ is the quotient of $S^3 \subset \mathbb{C}^2$ by the action of the group generated by  $\begin{pmatrix}\zeta & 0 \\ 0 & \zeta^q\end{pmatrix}$, where $\zeta$ is a primitive $p^{\rm th}$ root of $1$, which is also the link of the cyclic quotient singularity of type $\frac1p(1,q)$. With this description, $L(p,q)$ is obtained by doing $-p/q$-surgery on the unknot in $S^3$.

\subsection*{Organisation of the paper}
In Section~\ref{s:balls} we give different descriptions of the balls $B_{p,q}$.
In Section~\ref{s:2Fareytree} we introduce the 2-Farey tree and state some of its properties, which we use in Section~\ref{s:ANN} to prove Theorem~\ref{thm:ANN}.
In Section~\ref{s:compare} we put the results from Theorem~\ref{thm:ANN} in a common framework with those of Lisca and Parma~\cite{HD2}, and also give new recursive descriptions of the examples obtained in \cite{HD2}.
In Section~\ref{s:homotopyCP2s} we describe two constructions of embeddings of triples of rational balls, coming from plumbings, and one construction of embeddings of rational balls and pairs of rational balls, coming from knots with lens space surgeries.
Finally, in Section~\ref{s:data} we share and discuss some data.

\subsection*{Acknowledgements}
We would like to warmly thank Yank{\i} Lekili for a question which initiated this collaboration.
We also thank Nikolas Adaloglou, Giulia Carfora, Johannes Hauber, Woohyeok Jo, Paolo Lisca, Marco Marengon, Jongil Park, and Kyungbae Park for stimulating conversations.
We would like to thank Andrea Parma for providing helpful data and clarifications about his papers.
The first author thanks the University of Glasgow for their hospitality.
The second thanks Oxford University and Nantes University for their hospitality.
Part of this work has been carried out at several conferences and workshops, including \emph{Surfaces in 4-manifolds} in le Croisic, \emph{4-vari\'et\'es: \`a travers les dimensions}
at CIRM in Marseilles, \emph{The low-dimensional workshop} during the \emph{Singularities and low-dimensional topology} research semester at the Erd\H{o}s Center in Budapest and \emph{Combinatorial and gauge-theoretical methods in low-dimensional topology and geometry} at the Centro De Giorgi in Pisa.

The post-theorem introduction says: "The proof of Theorem~\ref{thm:ANN} is inductive, and relies on a tricky, but rather direct, Kirby calculus argument." More precisely, it "uses structural induction on a simple variant of the Farey tree" (to be defined in Section~\ref{s:2Fareytree}). The Kirby-calculus part is "related to Lisca and Parma's horizontal decompositions" and "informed by the Kirby diagrams of~\cite{EMPR}." The "main idea" is "to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\times D^3$."

The post-theorem introduction says: “The proof of the main theorem is inductive, and relies on a tricky, but rather direct, Kirby calculus argument.” More precisely, it “uses structural induction on a simple variant of the Farey tree” (to be defined next). The Kirby-calculus part is “related to Lisca and Parma's horizontal decompositions” and “informed by the Kirby diagrams of~\cite{EMPR}.” The “main idea” is “to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\times D^3$.”

Question:
Let \(B_{p,q}\) denote the paper’s rational homology ball associated to a pair of nonnegative integers \((p,q)\); in the cases relevant here it has boundary the lens space \(L(p^2,pq-1)\). Suppose \(p_1,p_2,q_1,q_2\) are nonnegative integers such that \(q_1\) and \(q_2\) are even, \(p_i\ge q_i\) for \(i=1,2\), and
\[
p_1q_2-p_2q_1=\pm 2.
\]
Which of the following conclusions about smooth embeddings into \(\overline{\mathbb{CP}}^{\,2}\) (the complex projective plane with reversed orientation) holds?

The disjoint union
\[
B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}
\]
admits a smooth orientation-preserving embedding in \(\overline{\mathbb{CP}}^{\,2}\).

Choices:
B.
The disjoint union
\[
B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}
\]
admits a smooth orientation-preserving embedding in \(\overline{\mathbb{CP}}^{\,2}\) provided one also assumes that \(p_1\) and \(p_2\) are coprime.
C.
There exists a smooth embedding in \(\overline{\mathbb{CP}}^{\,2}\) of the disjoint union
\[
B_{p_1,q_1}\sqcup B_{p_2,q_2}.
\]
D.
The disjoint union
\[
B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}
\]
admits a smooth orientation-preserving embedding in \(\overline{\mathbb{CP}}^{\,2}\) whenever \(q_1\) and \(q_2\) are even, \(p_i\ge q_i\), and \(p_1q_2-p_2q_1=\pm 1\).
E.
For every such choice of \(p_1,p_2,q_1,q_2\), there is a smooth orientation-preserving embedding in \(\overline{\mathbb{CP}}^{\,2}\) of
\[
B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{r,s}
\]
for some pair of nonnegative integers \((r,s)\) depending on \((p_1,p_2,q_1,q_2)\), not necessarily equal to \((p_1+p_2,q_1+q_2)\).

added unnecessary coprimality hypothesis on p_1,p_2

dropped the third summand B_{p_1+p_2,q_1+q_2}

Farey-determinant condition ±2 replaced by ±1

specific third ball determined by Farey sum replaced by unspecified existential third summand

The stem gives hypotheses and asks which conclusion holds, but it does not explicitly reveal the correct conclusion. The correct option is not signaled by wording in the stem.

The correct choice is essentially the theorem statement itself under the stated hypotheses. The item mainly tests recognition of the exact result rather than deriving a conclusion from broader principles.

There is some need to compare nearby variants of the theorem (extra hypothesis, weakened conclusion, altered determinant condition, existential variant), but the task is still mostly theorem recall rather than substantial generative reasoning.

The distractors are plausible and mathematically meaningful: one adds an unnecessary coprimality assumption, one gives a weaker true-looking statement, one alters the determinant condition, and one replaces the specific third summand by an existential claim. These reflect realistic confusion points.

A solid recall-based MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of the theorem and only moderately tests reasoning.