Let $\Omega \subset \mathbb{C}^N$ be a strictly pseudoconvex domain. Then every b-pluripolar $F_\sigma$ set of topological dimension $N$ must propagate into the interior.

\begin{theorem}\label{hausdorff}
   Every set $A\subset P_\beta$ of zero $\beta$-dimensional Hausdorff measure is non-propagating and b-pluripolar.
\end{theorem}

\begin{theorem}[Hedenmalm~\cite{hedenmalm}]\label{hedenmalmsats}
Let $\mathbb{B}$ denote the unit ball in $\C^N$, $N>1$, and let $\bar 0 \in \mathbb{B}$ denote its center. Then the set $M_{\bar 0}(\partial \mathbb{B})\setminus J_{\bar 0}(\partial \mathbb{B})$ is non-empty.
\end{theorem}

Let $\Omega$ denote a bounded domain in $\mathbb{C}^N$, and let $A \subset \partial\Omega$ be a set for which there exists a negative plurisubharmonic (psh) function $u$ defined on $\Omega$ ($u\in \mathcal{PSH}^-(\Omega)$), not identically $-\infty$, with
\[\limsup_{z\rightarrow A} u(z) = -\infty.\]
Such \textit{boundary pluripolar sets} (b-pluripolar for short) were recently characterized as the exceptional sets for the Dirichlet problem for the complex Monge--Ampère operator ~\cite{nilsson2}. More precisely, under the assumption that $\Omega$ admits strong plurisubharmonic barriers, in the sense that we at every boundary point $\zeta \in \partial\Omega$ can find $u \in \mathcal{PSH}(\Omega)\cap C(\widebar \Omega)$ with
\[
u(\zeta)=0, \quad u<0 \text{ on } \widebar \Omega \setminus \{\zeta\},
\]
then the Dirichlet problem
\[
 \begin{cases}
 u \in L^\infty\cap\mathcal{PSH}(\Omega) \\
 \det(\frac{\partial^2 u}{\partial z_j \partial \bar z_k})=0 \\
 \lim_{z \rightarrow \zeta}u(z) = \varphi(\zeta) \qquad \forall \zeta \in \partial\Omega \setminus E_\varphi
 \end{cases}
\]
 is uniquely solvable (in the weak sense of Bedford--Taylor~\cite{bedford-taylor, bedford-taylor2}) if and only if $E_\varphi$ is b-pluripolar. Here, we assume that $\varphi\colon \partial \Omega \rightarrow \mathbb{R}$ is bounded, and continuous outside $E_\varphi$. Furthermore, the unique solution is continuous~\cite[Theorem~5.1]{nilsson} outside the closure of
 \[
 \hat E_\varphi \coloneqq\{z \in \widebar\Omega \mathrel{;} u\in\mathcal{PSH}^-(\Omega), \limsup_{w\rightarrow E_\varphi}u(w)=-\infty \implies u(z) =-\infty\},
 \]
the \textit{boundary pluripolar hull} of $E_\varphi$ (if the boundary pluripolar hull contains interior points, we say that the set is \textit{propagating}). Since discontinuity sets are necessarily $F_\sigma$, this motivates investigating necessary and sufficient criteria for when boundary pluripolar $F_\sigma$ sets propagate.

In the first part of the paper, we obtain the following result:

\section{Boundary pluripolar sets of dimension \texorpdfstring{$N$}{N} must propagate}
Our aim in this section is to prove that on strictly pseudoconvex domains in $\C^N$,
a non-propagating b-pluripolar $F_\sigma$ set $A$ with $\dim A \geq N$ cannot exist. The argument will be based on the following result, which was established using a cohomological argument in the proof of the main theorem in~\cite{stout}.
\begin{lemma}[Stout]\label{stout}
Let $\Omega\subset \C^N$ be a bounded convex domain, and suppose that $A \subset \partial \Omega$ is a compact set such that for every neighborhood $U$ (in $\partial \Omega$) containing $p\in A$, there is a real hyperplane $\Pi_U: L(z)=0$ such that
\[
p \in \{L(z)>0\}\cap \partial \Omega \subset U.
\]
If each $(A\cap \{L(z)>0\}) \cup (\Pi_U \cap \widebar \Omega)$ is polynomially convex, then $\dim A < N$.
\end{lemma}
\begin{figure}[h]
\centering
\tikzset{every picture/.style={line width=0.75pt}}

To show that $A\cup(\Pi\cap\widebar \Omega)$ is polynomially convex, assume that $z_0 \in \widebar \Omega \setminus(A \cup \Pi)$. Multiplying $L$ with a constant if necessary, we may assume that $L(z_0)>1$. The preceding paragraph implies that we may find $\tilde u \in \psh(\C^N)$ with
\[
\tilde u(z_0)>-1, \quad \tilde u <0 \text { on }\widebar \Omega,\quad \tilde u< -\max_{\zeta\in A}|L(\zeta)| \text { on }A.
\]
The psh function $\tilde u + L$ then satisfies
\[
\tilde u(z_0)+L(z_0)>0, \quad \tilde u+L< 0 \text { on }A \cup (\Pi\cap \widebar \Omega),
\]
which finishes the proof.
\end{proof}
\begin{remark}
The assumption that $\Omega$ is B-regular (which is automatic if $\Omega$ is strictly pseudoconvex) compensates for the fact that $A$ is a peak set in Stout's proof. Indeed, if $A$ is a peak set, we can instead apply Lemma~\ref{approx} to $u = \max(C_1\log|1-h_A|,C_2)$ with suitable constants $C_2 <0<C_1$.
\end{remark}
The following simple example shows that the converse is not true: There are polynomially convex, b-pluripolar sets which do propagate into the interior.
\begin{example}
Let $\mathbb{B}$ denote the ball in $\C^2$, and take an arc $\gamma \subset \{|z_2|=1\}$. This set is b-pluripolar since $\gamma \subset \{\log|z_1| =-\infty\}$, and polynomially convex by~\cite[Corollary~3.1.2]{stoutbok}. Let $u \in \psh^-(\mathbb{B})$ be a function such that $\limsup_{w\rightarrow \gamma}u(w)=-\infty$. Since $\gamma$ has positive arc length, it is not b-(pluri)polar on the disk $\{(0, z_2) \suchthat |z_2|<1\}$. It follows that $u(0,z_2) \equiv -\infty$; in other words, $\gamma$ propagates along $\{z_1=0\}\cap\mathbb B$.
\end{example}

With Lemma~\ref{polyconv} at hand, we are now ready to prove the main result in this section.
\begin{theorem}\label{topdim}
Let $\Omega\Subset \C^N$ be strictly pseudoconvex, and let $A\subset \partial \Omega$ be a boundary pluripolar $F_\sigma$ set with topological dimension greater than $N-1$. Then $\hat A \cap \Omega$ is non-empty.
\end{theorem}
\begin{proof}
Let us first do some standard reductions. If we for a contradiction assume the existence of a non-propagating b-pluripolar $F_\sigma$ set $A$ with $\dim A \geq N$, then
\[
A= \bigcup K_j
\]
with $K_j$ compact. This implies~\cite[Theorem~3.1.8]{Engelking} that there exists $j_0$ with $\dim K_{j_0}\geq N$. Pick a point $p\in K_{j_0}$ such that each neighborhood $U$ containing $p$ satisfies $\dim U\cap K_{j_0}\geq N$. Since $\Omega$ is strictly pseudoconvex, there exists an open set $V$ containing $p$ and a local biholomorphism $f$ such that $f(\partial \Omega\cap V)$ is strictly convex in a neighborhood of $K\coloneq f(U\cap K_{j_0})$. Also, if $A$ does not propagate, then $K$ does not propagate into $B(p,\epsilon)\cap f(\Omega\cap V)$. We may therefore assume that the b-pluripolar set $A$ is compact, and that $\Omega$ is convex and B-regular, and even strictly convex in a neighborhood of $A$.

\begin{definition}
Given a family of measures $\mathcal{M}\subset M(\partial\Omega)$, we say that a set $A\subset \partial \Omega$ is \textit{totally null} with respect to $\mathcal{M}$ if $\mu(A)=0$ for every $\mu \in \mathcal{M}$.
\end{definition}
It is well-known (using the Cole--Range theorem and Bishop's theorem, cf. for example~\cite[pp.~204--205]{Rudin}) that a compact set $K$ is totally null with respect to $M_{z_0}(\partial\Omega)$ if and only if $K$ is a \textit{zero set} for $A(\Omega)$, in other words there exists $f\in A(\Omega)$ such that $f(\zeta) = 0$ for every $\zeta \in K$ and $f(z) \neq 0$ for every $z \in \widebar \Omega \setminus K$. We can then construct
\[
u(z) \coloneq \log|f(z)| - \max_{w\in \widebar \Omega} \log|f(w)|
\]
which shows that $K$ is b-pluripolar and non-propagating. In particular, for any compact, propagating b-pluripolar set $\tilde K$ that does not propagate to $z_0$ (take for example $\tilde K \coloneq \partial \Omega \cap \Pi$, where $\Pi$ is a complex hyperplane not containing $z_0$), there exists $\nu \in M_{z_0}(\partial \Omega)$ such that $\nu (\tilde K) >0$. To generalize Theorem~\ref{hedenmalmsats}, it is therefore enough to prove
\begin{theorem}\label{jensen}
Let $A\subset \partial \Omega$ be a b-pluripolar set that does not propagate to $z_0$. Then $A$ is totally null with respect to $J_{z_0}(\partial \Omega)$.
\end{theorem}
\begin{proof}
Let $\psh^b(\Omega)\coloneq \psh(\Omega) \cap \usc(\widebar \Omega)$ denote the upper bounded psh functions on $\Omega$ (with upper semicontinuous extension to the boundary). We shall first show that for every $\mu \in J_{z_0}(\partial\Omega)$, we have
\[
u(z_0) \leq \int_{\partial \Omega} u \ d\mu, \quad u\in \psh^b(\Omega).
\]
By Wikström's approximation theorem and the monotone convergence theorem, it is enough to consider $u \in \psh(\Omega) \cap C(\widebar \Omega)$. As in the proof of Lemma~\ref{approx}, we can find $\Omega_j\Supset \Omega$ and $u_{j} \in \psh(\Omega_j)\cap C(\Omega_j)$ such that $|u-u_{j}|<\frac{1}{j}$ on $\widebar \Omega$. Furthermore, as $\Omega$ is strictly pseudoconvex, we can assume that $\Omega_j$ is strictly pseudoconvex as well. Reasoning as in the proof of \cite[Corollary~1.3.10]{stoutbok}, the approximation result of Bremermann--Sibony yields $f_1,\dots,f_n\in \mathcal{O}(\Omega_j) \subset A(\Omega)$ and constants $1\geq c_1,\dots, c_n>0$ such that
\[
u_j - \frac{1}{j}\leq \max_{k=1,\dots,n} c_k \log |f_k| \leq u_j
\]
on $\widebar \Omega$. Consequently, there is a $f_{k_0}$ such that
\[
u_j(z_0) - \frac{1}{j} \leq c_{k_0} \log |f_{k_0}(z_0)| \leq u_j(z_0).
\]
Hence,
\[
u_j(z_0) - \frac{1}{j}\leq c_{k_0}\log|f_{k_0}(z_0)|\leq \int_{\partial \Omega}\max_{k=1,\dots,n} c_k \log |f_k| \ d\mu \leq\int_{\partial \Omega} u_j \ d\mu,
\]
and so the assertion follows by letting $j\rightarrow \infty$. In the special case that $u_A$ is a negative psh function with $u_A(z_0)>-\infty$ and $u_A(\zeta) =-\infty$ for every $\zeta \in A$, then
\[
 -\infty< \int_{\partial \Omega }u_A \ d\mu\leq \int_Au_A \ d\mu.
\]
We conclude that $\mu(A)=0$, which finishes the proof.
\end{proof}
\begin{corollary}\label{hedenmalmgen} Let $\Omega\subset \C^N, N>1$ be a smooth, strictly pseudoconvex domain. Then for every $z_0 \in \Omega$, $M_{z_0}(\partial \Omega)\setminus J_{z_0}(\partial \Omega)$ is non-empty.
\end{corollary}
\begin{remark}
The assumption that $\Omega$ is smooth and strictly pseudoconvex can be relaxed somewhat, since the Cole--Range theorem holds on any domain where $A(\Omega)$ is \textit{tight}. See~\cite{boos} and the references therein for more details.
\end{remark}
\bibliographystyle{amsplain}
\bibliography{name}

\noindent
Contrapositively, non-propagating b-pluripolar sets have at most dimension $N-1$. The proof is essentially an adaptation of an argument due to Stout~\cite{stout}, who proved the corresponding statement for peak sets, i.e. closed sets $K \subset \partial \Omega$ for which there exists $h_K\in A(\Omega)$ (holomorphic on $\Omega$ and continuous on the closure) with
\[
h_K = 1 \text{ on }K, \quad |h_K| < 1 \text{ on }\widebar \Omega \setminus K.
\]
Roughly, the idea is to show that even under the weaker assumption that $K$ is non-propagating and b-pluripolar, $K$ is still polynomially convex. The final step in Stout's proof
then implies that $K$ has some topological properties which are incompatible with $\dim K \geq N$.

In Section~3, we establish sufficient criteria for b-pluripolarity in terms of Hausdorff dimension. Given a bounded domain $\Omega$ and a parameter $\beta>0$, we consider the set $P_\beta$ consisting of $\zeta \in \partial \Omega$ for which there exists a peak function $h_{\{\zeta\}}$ with
\[
\abs{h_{\{\zeta\}}(z)-1} \leq C_\zeta \|z-\zeta\|^\beta,
\]
where $\| \cdot\|$ denotes the Euclidean norm. We show
\begin{theoremB}
Every set $A\subset P_\beta$ of zero $\beta$-dimensional Hausdorff measure is non-propagating and b-pluripolar.
\end{theoremB}
\noindent
 The argument is quite elementary; we simply postcompose peak functions with the harmonic measure of arcs on the circle.

Finally, in Section~4, we compare representing measures and Jensen measures at $z_0 \in \Omega$ supported on $\partial \Omega$, where $\Omega$ is smooth and strictly pseudoconvex. We show that the compact totally null sets of the two classes differ; the former are the zero sets for $A(\Omega)$, in particular b-pluripolar and non-propagating, while the class of totally null sets for Jensen measures contains every b-pluripolar set that does not propagate to $z_0$. From this, we deduce
\begin{theoremC}
Let $\Omega\subset \mathbb{C}^N$, $N>1$ be a smooth, strictly pseudoconvex domain. Then there are always representing measures that are not Jensen measures.
\end{theoremC}
\par \noindent
 This extends a result of Hedenmalm (\cite[Theorem~3.7]{hedenmalm}, see also Theorem~\ref{hedenmalmsats} below) on the unit ball.

We have labeled the main results as Theorems A--C for ease of reference;
these correspond to Theorems~\ref{topdim}, \ref{hausdorff}, and Corollary~\ref{hedenmalmgen}.

Contrapositively, the introduction states that "non-propagating b-pluripolar sets have at most dimension $N-1$." The proof is described as "essentially an adaptation of an argument due to Stout" for peak sets. "Roughly, the idea is to show that even under the weaker assumption that $K$ is non-propagating and b-pluripolar, $K$ is still polynomially convex." Then "the final step in Stout's proof then implies that $K$ has some topological properties which are incompatible with $\dim K \geq N$."

Contrapositively, the introduction states that non-propagating b-pluripolar sets have at most dimension $N-1$. The proof is described as essentially an adaptation of an argument due to Stout for peak sets. Roughly, the idea is to show that even under the weaker assumption that $K$ is non-propagating and b-pluripolar, $K$ is still polynomially convex. Then the final step in Stout's proof implies that $K$ has some topological properties which are incompatible with $\dim K \geq N$.

Question:
Let \(\Omega\subset \mathbb{C}^N\) be a strictly pseudoconvex domain, and let \(A\subset \partial\Omega\) be an \(F_\sigma\) set of topological dimension \(N\). Assume that \(A\) is boundary pluripolar, meaning that there exists a negative plurisubharmonic function \(u\in \mathcal{PSH}^-(\Omega)\), not identically \(-\infty\), such that \(\limsup_{z\to A}u(z)=-\infty\). Its boundary pluripolar hull is
\[
\hat A=\{z\in \overline\Omega: ყოვall\ v\in \mathcal{PSH}^-(\Omega),\ \limsup_{w\to A}v(w)=-\infty \implies v(z)=-\infty\}.
\]
Which statement holds?

Such a set \(A\) must propagate into the interior; equivalently, its boundary pluripolar hull meets the interior of \(\Omega\), i.e. \(\hat A\cap \Omega\neq \varnothing\).

Choices:
B. Such a set \(A\) need not propagate into the interior; it is possible that \(\hat A\cap \Omega=\varnothing\), and in that case one can still conclude only that \(A\) is polynomially convex in \(\partial\Omega\).
C. Such a set \(A\) cannot have empty boundary pluripolar hull in the sense that \(\hat A\neq A\); in particular, the hull contains at least one point of \(\overline\Omega\setminus A\).
D. Such a set \(A\) must propagate into the interior only when its topological dimension is strictly greater than \(N\); dimension exactly \(N\) does not force \(\hat A\cap \Omega\neq \varnothing\).
E. For every strictly pseudoconvex domain \(\Omega\subset\mathbb C^N\), there exists a point \(z_0\in \Omega\) such that every boundary pluripolar \(F_\sigma\) set \(A\subset\partial\Omega\) of topological dimension \(N\) propagates to that same point, i.e. \(z_0\in \hat A\).

dropped the conclusion that the hull meets the interior

existential interior propagation point made uniform in A


\label{thm main}
Assume that for $1 \leq i \leq N$, the mean curvatures $H_{i, \mp}$ are strictly positive, and that there exists some $1 \leq \Lambda < N$ such that
\begin{align*}
\underline H_{i,-} \leq \overline H_{i,+} & \text{for } & 1 \leq i \leq \Lambda , \\
\underline H_{i,-} \geq \overline H_{i,+} & \text{for } & \Lambda + 1\leq i \leq N.
\end{align*}
Furthermore, assume that
\begin{align}\label{eq:condition-on-squares-of-mean-curvatures}
    \sum_{1 \leq i \leq N} \left(\underline H_{i,-}^2 - \overline H_{i,+}^2\right) \ge 0.
\end{align}
Then the ADM mass of $( M,g)$ is nonnegative.

Extremality and rigidity results in scalar curvature geometry on smooth Riemannian manifolds often remain valid for Riemannian metrics with singularities, provided that the relevant curvature restrictions are still satisfied in a weak sense.
To give a well-known example, the positive mass theorem for asymptotically flat manifolds holds for manifolds with metric singularities along hypersurfaces, if the singularities are {\em mean-convex}, that is, the mean curvature jump with respect to the infinity pointing normal vector is non-positive, see \cite{shi-tam-manifolds-with-boundary, Miao2002cornersATP}.
As described in \cite[Proposition 3.1]{Miao2002cornersATP}, {\em mean-concave} singularities (with a strictly negative mean curvature jump) can be regarded as having infinitely negative scalar curvature in a weak sense.

\medskip

This paper shows that the above mean-convexity condition can be replaced with a weaker one: The effect of mean-concave jumps at some components of the singular set can be offset by mean-convex jumps at others.
Loosely speaking, the {\em local} mean-convexity condition can be replaced by a weaker {\em global} one by passing to some average.
This phenomenon is unexpected, given that the pointwise lower scalar curvature bound in the positive mass theorem cannot be replaced by a lower bound on an averaged scalar curvature.

\medskip

To state our main result, let $M$ be a non-compact connected spin manifold which is decomposed along smooth compact hypersurfaces $\Sigma_1,\dots\Sigma_N$ as
\[
M = M_0 \cup_{\Sigma_1} \dots\cup_{\Sigma_N} M_N.
\]
Here $M_0, \dots, M_N \subset M$ are smooth submanifolds with boundary, $M_{i}$ is compact for $1 \leq i \leq N-1$, and $M_N$ is noncompact and connected; see \cref{fig:corners}.
For $0 \leq i \leq N$, let $g_i$ be smooth Riemannian metrics on $M_i$ such that
\begin{itemize}
    \item for $1 \leq i \leq N$, we have $(g_{i-1})|_{\Sigma_i} = (g_i)|_{\Sigma_i}$,
    \item each $g_i$ has non-negative scalar curvature,
    \item $(M_{N},g_N)$ is asymptotically flat.
\end{itemize}
We write $g\coloneqq g_0\cup\dots\cup g_N$ for the induced $C^0$-Riemannian metric on $M$.
Let us denote the mean curvatures along $\Sigma_i$ by
\[H_{i,-}\coloneqq H(\Sigma_i\subseteq M_{i-1}), \qquad H_{i,+}\coloneqq H(\Sigma_i\subseteq M_{i}), \qquad 1 \leq i \leq N.
\]
We adopt the convention common in general relativity that mean curvatures are computed with respect to the infinity-pointing normal.
That is, if $M$ is the Euclidean space $\mathbb{R}^{n}$ with the decomposition
\[
    M_0 := \{ |x| \leq 1\}, \quad M_1:=\{1 \leq |x| \leq 2\}, \quad M_2 := \{|x| \geq 2\},
\]
then we have $H_{1,\mp} = n-1,\ H_{2,\mp } = (n-1)/2$.

\medskip

For $1 \leq i \leq N$, we set
\begin{align*}
    \underline H_{i,-} := \min_{\Sigma_i} H_{i, -}, \qquad \overline H_{i,+} := \max_{\Sigma_i} H_{i, +}.
\end{align*}
Our main result is the following.

Extremality and rigidity results in scalar curvature geometry on smooth Riemannian manifolds often remain valid for Riemannian metrics with singularities, provided that the relevant curvature restrictions are still satisfied in a weak sense.
To give a well-known example, the positive mass theorem for asymptotically flat manifolds holds for manifolds with metric singularities along hypersurfaces, if the singularities are {\em mean-convex}, that is, the mean curvature jump with respect to the infinity pointing normal vector is non-positive, see \cite{shi-tam-manifolds-with-boundary, Miao2002cornersATP}.
As described in \cite[Proposition 3.1]{Miao2002cornersATP}, {\em mean-concave} singularities (with a strictly negative mean curvature jump) can be regarded as having infinitely negative scalar curvature in a weak sense.

This paper shows that the above mean-convexity condition can be replaced with a weaker one: The effect of mean-concave jumps at some components of the singular set can be offset by mean-convex jumps at others.
Loosely speaking, the {\em local} mean-convexity condition can be replaced by a weaker {\em global} one by passing to some average.
This phenomenon is unexpected, given that the pointwise lower scalar curvature bound in the positive mass theorem cannot be replaced by a lower bound on an averaged scalar curvature.

To state our main result, let $M$ be a non-compact connected spin manifold which is decomposed along smooth compact hypersurfaces $\Sigma_1,\dots\Sigma_N$ as
\[
M = M_0 \cup_{\Sigma_1} \dots\cup_{\Sigma_N} M_N.
\]
Here $M_0, \dots, M_N \subset M$ are smooth submanifolds with boundary, $M_{i}$ is compact for $1 \leq i \leq N-1$, and $M_N$ is noncompact and connected; see \cref{fig:corners}.
For $0 \leq i \leq N$, let $g_i$ be smooth Riemannian metrics on $M_i$ such that
\begin{itemize}
 \item for $1 \leq i \leq N$, we have $(g_{i-1})|_{\Sigma_i} = (g_i)|_{\Sigma_i}$,
 \item each $g_i$ has non-negative scalar curvature,
 \item $(M_{N},g_N)$ is asymptotically flat.
\end{itemize}
We write $g\coloneqq g_0\cup\dots\cup g_N$ for the induced $C^0$-Riemannian metric on $M$.
Let us denote the mean curvatures along $\Sigma_i$ by
\[H_{i,-}\coloneqq H(\Sigma_i\subseteq M_{i-1}), \qquad H_{i,+}\coloneqq H(\Sigma_i\subseteq M_{i}), \qquad 1 \leq i \leq N.
\]
We adopt the convention common in general relativity that mean curvatures are computed with respect to the infinity-pointing normal.
That is, if $M$ is the Euclidean space $\mathbb{R}^{n}$ with the decomposition
\[
 M_0 := \{ |x| \leq 1\}, \quad M_1:=\{1 \leq |x| \leq 2\}, \quad M_2 := \{|x| \geq 2\},
\]
then we have $H_{1,\mp} = n-1,\ H_{2,\mp } = (n-1)/2$.

\medskip

For $1 \leq i \leq N$, we set
\begin{align*}
 \underline H_{i,-} := \min_{\Sigma_i} H_{i, -}, \qquad \overline H_{i,+} := \max_{\Sigma_i} H_{i, +}.
\end{align*}
Our main result is the following.

\label{fig:corners}
\end{figure}

For the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.
When chosen appropriately, the te

For $1 \leq i \leq N$, we set
\begin{align*}
 \underline H_{i,-} := \min_{\Sigma_i} H_{i, -}, \qquad \overline H_{i,+} := \max_{\Sigma_i} H_{i, +}.
\end{align*}
Our main result is the following.

\begin{lemma}\label{lemma main}
 Suppose that $\Sigma$ is a smooth $(n-1)$-manifold and that $H_-,H_+\colon \Sigma\to\bbR$ are smooth functions, and let $\underline H_{-},\overline H_+\in(0,\infty)$ be such that $H_-\ge \underline H_-$ and $H_+\le\overline H_+$.
 Given $a\in\bbR$ satisfying $\overline{H}_+^2-\underline{H}_-^2+a^2\ge0$, we define functions $\vec{H}_+, \vec{H}_- \colon \Sigma \to \mathbb{R}^{1,1}$ via
 \begin{align*}
 \vec H_-=\begin{pmatrix}
 H_-\\
 a
 \end{pmatrix},
 \qquad
 \vec H_+=\begin{pmatrix}
 H_+\\
 \sqrt{\overline H_+^2-\underline H_-^2+a^2}
 \end{pmatrix}.
 \end{align*}
 Then there exists a hyperbolic rotation angle $\vartheta\in \mathbb R$ such that the function
 \begin{align*}
 X=F_\vartheta \vec H_--\vec H_+ \colon \Sigma \to \mathbb R^{1,1}, \qquad F_\vartheta=
 \begin{pmatrix}
 \cosh \vartheta & -\sinh \vartheta\\
 -\sinh \vartheta &\cosh\vartheta
 \end{pmatrix},
 \end{align*}
 satisfies
 \begin{align*}
 X_1\ge|X_2| \quad \textrm{ along } \Sigma,
 \end{align*}
 where $X_1$ and $X_2$ denotes the first and second component of $X$, respectively.
\end{lemma}

\begin{proof}
Let us abbreviate $\xi\coloneqq\sqrt{\overline H_+^{2}-\underline H_-^{2}+a^2}$.
We choose
\begin{align}\label{eq vartheta 1}
 \vartheta =\sinh^{-1}\left(
 \frac{a\overline H_+ - \xi\underline H_-}
 {\underline H_-^{2}-a^2}
 \right),
\end{align}
in case $\underline H_-\ne a$, and
\begin{align}\label{eq vartheta 2}
 \vartheta=\log(\underline H_-)-\log(\overline H_+)
\end{align}
in case $\underline H_-=a$.
Using $\cosh(\vartheta) = \sqrt{1+\sinh^2(\vartheta)}$, we obtain
\begin{align*}
 & F_\vartheta \vec H_- - \vec H_+
 =
 \begin{pmatrix}
 \sqrt{1+\sinh^2(\vartheta)}\,H_- - \sinh(\vartheta)\, a - H_+\\[0.4em]
 \sqrt{1+\sinh^2(\vartheta)}\,a - \sinh(\vartheta)\, H_-
 - \xi
 \end{pmatrix}.
\end{align*}
If $\underline H_-\neq a$, then
\begin{align*}
 &\left(\underline H_-^{2}-a^2\right)^2\left(1+\sinh^2(\vartheta)\right)\\
 &\qquad\qquad = \underline H_-^4 - 2a^2\underline H_-^2 + a^4 + (a\overline H_+ - \xi\underline H_-)^2\\
 &\qquad\qquad = \underline H_-^4 - 2a^2\underline H_-^2 + a^4 + a^2\overline H_+^2 - 2a\xi\overline H_+\underline H_- + (\overline H_+^2-\underline{H}_-^2+a^2)\underline H_-^2\\
 &\qquad\qquad = \left(\overline H_+ \underline H_- - a\xi\right)^2
\end{align*}
and hence $\left|\underline H_-^{2}-a^2\right|\sqrt{1+\sinh^2(\vartheta)}=\left|\overline H_+\underline H_- - a \xi\right|$.

In the case that $\underline H_->a$, we note that $\xi< \overline H_+$, and that both $\underline H_-^{2} -a^2$ and $\overline H_+ \underline{H}_--a\xi$ are positive.
We can thus estimate
\begin{align*}
 &(\underline H_-^{2} - a^2)
 \left(\sqrt{1+\sinh^2(\vartheta)}\,H_- - \sinh(\vartheta)\, a - H_+\right)\\
 &\qquad\qquad= H_- \left(\overline H_+ \underline H_- - a \xi \right) - a \left(
 a\overline H_+ - \xi\underline H_-\right)- H_+(\underline H_-^{2}-a^2)\\
 &\qquad\qquad\ge(H_- -\underline H_-)\left(\overline H_+ \underline H_- - a \xi\right),
\end{align*}
where we used $H_+(\underline H_-^{2}-a^2)\le \overline H_+(\underline H_-^{2}-a^2)$ for the final inequality.
Moreover, we have
\begin{align*}
 &(\underline H_-^{2}-a^2)\left(\sqrt{1+\sinh^2(\vartheta)}\,a - \sinh(\vartheta)\, H_- - \xi\right)\\
 &\qquad\qquad= a \left(\overline H_+ \underline H_- - a \xi\right) - H_- \left(a\overline H_+ - \xi\underline H_-\right)- \xi(\underline H_-^{2}-a^2)\\
 &\qquad\qquad=(H_--\underline H_-)\left(\xi\underline H_--a\overline H_+\right)
\end{align*}
Since
\begin{align}\label{computation1}
\begin{split}
 &\left(\overline H_+ \underline H_- - a \xi \right)^2 -\left(\xi\underline H_- -a\overline H_+\right)^2\\
 &\qquad=\overline H_+^2\underline H_-^2- 2a\xi\overline H_+\underline H_- + a^2\xi^2 - \underline H_-^2\xi^2 + 2a\xi\overline H_+\underline H_- - a^2\overline H_+^2\\
 &\qquad= \overline H_+^2\underline H_-^2 + a^2 \left(\overline H_+^{2}-\underline H_-^{2}+a^2\right) - \underline H_-^2\left(\overline H_+^{2}-\underline H_-^{2}+a^2\right)- a^2 \overline H_+^2\\
 &\qquad =a^4-2a^2\underline H_-^2+\underline H_-^4 \ge0,
 \end{split}
\end{align}
we obtain $\overline H_+ \underline H_- - a\xi\ge |\xi\underline H_--a\overline H_+|$ and consequently,
\begin{align}\label{computation2}
 \begin{split}
 &\left(\sqrt{1+\sinh^2(\vartheta)}\,H_- - \sinh(\vartheta)\, a - H_+\right)\\
 &\qquad\qquad\qquad\ge \Big|\sqrt{1+\sinh^2(\vartheta)}\,a - \sinh(\vartheta)\, H_- - \xi\Big|.
 \end{split}
\end{align}
In the case $\underline H_-<a$, where $\xi> \overline H_+$ and both $\underline H_-^{2} - a^2$ and $\overline H_+ \underline{H}_--a\xi$ are negative, we can similarly estimate
\begin{align*}
 &-(\underline H_-^{2} - a^2)
 \left(\sqrt{1+\sinh^2(\vartheta)}\,H_- - \sinh(\vartheta)\, a - H_+\right)\\
 &\qquad\qquad= -H_- \left(\overline H_+ \underline H_- - a \xi \right) + a \left(
 a\overline H_+ - \xi\underline H_-\right)+ H_+(\underline H_-^{2}-a^2)\\
 &\qquad\qquad\ge (H_- -\underline H_-)\left(-\overline H_+ \underline H_- + a \xi \right)\\
 &\qquad\qquad\ge (H_- -\underline H_-)\left|a \overline H_+ - \xi\underline H_- \right|,
\end{align*}
where the last inequality follows as in \eqref{computation1}.
Moreover,
\begin{align*}
 &-(\underline H_-^{2}-a^2)\left(\sqrt{1+\sinh^2(\vartheta)}\,a - \sinh(\vartheta)\, H_- - \xi\right)\\
 &\qquad\qquad= -a \left(\overline H_+ \underline H_- - a \xi\right) + H_- \left(a\overline H_+ - \xi\underline H_-\right)+ \xi(\underline H_-^{2}-a^2)\\
 &\qquad\qquad=(H_--\underline H_-)\left(a\overline H_+-\xi\underline H_-\right)
\end{align*}
Consequently, \eqref{computation2} holds in this case as well.

\begin{proof}[Proof of \cref{thm main}]
 Put $d_0=c_0= 0$, and for $1 \leq \ell \leq N$, put
 \begin{align}\label{eq:def-of-c-and-d}
 d_\ell :=\sum_{i=1}^\ell(\overline H_{i,+}^2-\underline H_{i,-}^2), \quad c_{\ell} := \sqrt{ \max \{d_\ell, 0\}}.
 \end{align}
 Without loss of generality, we can assume that $d_{\ell} > 0$ for some $\ell$.
 Otherwise, \cref{thm main} is implied by \cite{Miao2002cornersATP}.

Let $\ell \in \{ 1, \ldots, \Lambda' \}$.
 If we choose
 \begin{align}\label{eq:choice-for-lemma}
 \Sigma\coloneqq\Sigma_\ell,\quad H_\mp := H_{\ell,\mp},\quad \underline H_-\coloneqq \underline H_{\ell,-},\quad \overline H_+\coloneqq \overline H_{\ell,+}\quad\text{and}\quad a:=c_{\ell-1}
 \end{align}
 then $c_\ell = \sqrt{d_\ell} = \sqrt{\overline H_+^2-\underline H_-^2+a^2}$, and we obtain from \cref{lemma main} a hyperbolic angle $\vartheta_\ell$ such that
 \begin{align*}
 X=F_{\vartheta_\ell}\begin{pmatrix}
 H_{\ell,-}\\ c_{\ell-1}
 \end{pmatrix}
 -\begin{pmatrix}
 H_{\ell,+}\\ c_\ell
 \end{pmatrix}\in C^\infty\left(\Sigma_\ell,\mathbb{R}^{1,1}\right)
 \end{align*}
 satisfies $X_1\ge|X_2|$.

For $\ell \in \{ \Lambda' +1, \ldots, N \}$, we choose $\Sigma$, $H_-$, $H_+$, $\underline H_-$, and the constant $a$ as in \eqref{eq:choice-for-lemma}, and we choose
 \[\overline H_+ := \sqrt{ \underline H_{\ell,-}^2-c^2_{\ell-1}}.\]
 Note, that by the assumption on $\Lambda'$, we have $\underline H_{\ell,-}^2-c^2_{\ell-1}\ge \overline H_{\ell,+}^2$ and hence $\overline H_+ \geq \overline H_{\ell,+}\ge H_+$ and $\overline H_+^2 - \underline H_-^2 + a^2 = 0$.
 Therefore, an application of \cref{lemma main} gives a $\vartheta\in\bbR$ such that
 \begin{align*}
 X=F_{\vartheta_\ell}\begin{pmatrix}
 H_{\ell,-}\\ c_{\ell-1}
 \end{pmatrix}
 -\begin{pmatrix}
 H_{\ell,+} \\ 0
 \end{pmatrix}\in C^\infty\left(\Sigma_\ell,\mathbb{R}^{1,1}\right)
 \end{align*}
 again satisfies $X_1\ge|X_2|$.

\begin{figure}[ht]
    \begin{tikzpicture}
        \node at (0,0) {\includegraphics[width=.8\textwidth]{graphics/PDF/four-creases-with-locks.pdf}};
        \node[anchor=west] at (-6,3) {$ M = M_0\cup\dots\cup M_4$:};
        \node[scale=0.8] at (-4,2.1) {$M_0$};
        \node[scale=0.8] at (-2,2.3) {$M_1$};
        \node[scale=0.8] at (1,2.5) {$M_2$};
        \node[scale=0.8] at (2.9,1.9) {$M_3$};
        \node[scale=0.8] at (4.5,3) {$M_4$};
        \node[scale=0.8] at (-2.3,0.1) {$\Sigma_1$};
        \node[scale=0.8] at (-0.2,0.1) {$\Sigma_2$};
        \node[scale=0.8] at (2.6,0.1) {$\Sigma_3$};
        \node[scale=0.8] at (4.3,0.1) {$\Sigma_4$};

        \draw[stealth-stealth](-2.16,-3.63) to (-2.16,-2.7);
        \draw[stealth-stealth](-.13,-3.63) to (-.13,-2);
        \draw[stealth-stealth](2.74,-3.63) to (2.74,-2.4);

        \node[anchor=west] at (-6,-2) {$k=\frac{c_i}{n-1}\cdot g$:};
        \node[scale=0.8] at (-4,-3.2) {$c_0=0$};
        \node[scale=0.8] at (-1.81,-3.2) {$\frac{c_1}{n-1}$};
        \node[scale=0.8] at (0.22,-3.2) {$\frac{c_2}{n-1}$};
        \node[scale=0.8]at (3.09,-3.2) {$\frac{c_3}{n-1}$};
        \node[scale=0.8] at (4.5,-3.2) {$c_4=0$};
    \end{tikzpicture}
    \caption{The lock principle for $N=4$ and $\Lambda=2$.
    Two mean-concave curvature jumps at $\Sigma_1$ and $\Sigma_2$ are offset by two mean-convex curvature jumps at $\Sigma_3$ and $\Sigma_4$.
    The constants $c_i$ determining the symmetric tensor $k$ are defined in \eqref{eq:def-of-c-and-d}.}\label{fig:corners}
\end{figure}

For the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.
When chosen appropriately, the tensor $k$ acts as a \emph{lock}: it allows one to convert an unfavorable mean curvature jump at $\Sigma_i$ into a corresponding jump of $k$, to transport this defect across $ M_i$, and finally to absorb it at a later singularity $\Sigma_{j}$ with a favorable mean curvature jump.
A typical situation is shown in \cref{fig:corners}.
In this situation, we can then apply the positive mass theorem for initial data sets with creases proven by Kazaras--Khuri--Lin \cite{KazarasKhuriLin2025}.
By moving to initial data sets, we thus prove a result in Riemannian geometry that cannot be achieved by local deformation arguments.

\medskip

This paper elaborates on a particular example of the lock principle.
In \cref{sec extensions}, we discuss possible extensions.

\medskip

\noindent \textbf{Acknowledgements:} BH and SH are grateful to the Lonavala Geometry Festival for the ideal working conditions where part of this work was carried out.
BH was partially supported by the DFG-SPP 2026 ``Geometry at Infinity''.

For the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$. When chosen appropriately, the tensor $k$ acts as a \emph{lock}: it allows one to convert an unfavorable mean curvature jump at $\Sigma_i$ into a corresponding jump of $k$, to transport this defect across $ M_i$, and finally to absorb it at a later singularity $\Sigma_{j}$ with a favorable mean curvature jump (a typical situation is shown in \cref{fig:corners}). In this situation, one can then apply the positive mass theorem for initial data sets with creases proven by Kazaras--Khuri--Lin \cite{KazarasKhuriLin2025}. By moving to initial data sets, the argument proves a Riemannian result that \emph{cannot be achieved by local deformation arguments}.

For the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$. When chosen appropriately, the tensor $k$ acts as a \emph{lock}: it allows one to convert an unfavorable mean curvature jump at $\Sigma_i$ into a corresponding jump of $k$, to transport this defect across $ M_i$, and finally to absorb it at a later singularity $\Sigma_{j}$ with a favorable mean curvature jump (a typical situation is shown in
\label{fig:corners}
\end{figure}

For the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.
When chosen appropriately, the te}). In this situation, one can then apply the positive mass theorem for initial data sets with creases proven by Kazaras--Khuri--Lin \cite{KazarasKhuriLin2025}. By moving to initial data sets, the argument proves a Riemannian result that \emph{cannot be achieved by local deformation arguments}.,

Question:
Let $M$ be a non-compact connected spin manifold decomposed along smooth compact hypersurfaces $\Sigma_1,\dots,\Sigma_N$ as
\[
M=M_0\cup_{\Sigma_1}\cdots\cup_{\Sigma_N} M_N,
\]
where $M_0,\dots,M_N$ are smooth submanifolds with boundary, $M_i$ is compact for $1\le i\le N-1$, and $M_N$ is noncompact and connected. For each $0\le i\le N$, let $g_i$ be a smooth Riemannian metric on $M_i$ such that for every $1\le i\le N$ the induced metrics agree on the interface, $(g_{i-1})|_{\Sigma_i}=(g_i)|_{\Sigma_i}$, each $g_i$ has nonnegative scalar curvature, and $(M_N,g_N)$ is asymptotically flat. Let $g:=g_0\cup\cdots\cup g_N$ be the resulting $C^0$ Riemannian metric on $M$.

For each interface $\Sigma_i$, define
\[
H_{i,-}:=H(\Sigma_i\subset M_{i-1}),\qquad H_{i,+}:=H(\Sigma_i\subset M_i),
\]
where mean curvature is computed with respect to the infinity-pointing normal, and set
\[
\underline H_{i,-}:=\min_{\Sigma_i} H_{i,-},\qquad \overline H_{i,+}:=\max_{\Sigma_i} H_{i,+}.
\]
Assume that for every $1\le i\le N$, both $H_{i,-}$ and $H_{i,+}$ are strictly positive functions. Assume also that there exists an index $\Lambda$ with $1\le \Lambda < N$ such that
\[
\underline H_{i,-}\le \overline H_{i,+}\quad \text{for }1\le i\le \Lambda,
\qquad
\underline H_{i,-}\ge \overline H_{i,+}\quad \text{for }\Lambda+1\le i\le N,
\]
and that
\[
\sum_{i=1}^N\bigl(\underline H_{i,-}^2-\overline H_{i,+}^2\bigr)\ge 0.
\]
Under these hypotheses, which conclusion about the standard ADM mass of the asymptotically flat manifold $(M,g)$ holds?

The ADM mass of $(M,g)$ is nonnegative.

Choices:
B.
The ADM mass of $(M,g)$ is nonnegative provided one only assumes
\[
\sum_{i=1}^N\bigl(\underline H_{i,-}-\overline H_{i,+}\bigr)\ge 0,
\]
while keeping all the other hypotheses unchanged.
C. There exists at least one asymptotically flat end of $(M,g)$ whose ADM mass is nonnegative.
D.
The ADM mass of $(M,g)$ is nonnegative even without requiring the existence of an index $\Lambda$ for which
\[
\underline H_{i,-}\le \overline H_{i,+}\quad (1\le i\le \Lambda),
\qquad
\underline H_{i,-}\ge \overline H_{i,+}\quad (\Lambda+1\le i\le N),
\]
as long as the scalar curvature assumptions and
\[
\sum_{i=1}^N\bigl(\underline H_{i,-}^2-\overline H_{i,+}^2\bigr)\ge 0
\]
still hold.
E. The ADM mass of $(M,g)$ is strictly positive unless one has $H_{i,-}=H_{i,+}$ on every interface $\Sigma_i$ for all $1\le i\le N$.

square-sum balance replaced by linear average

global conclusion on the ADM mass of $(M,g)$ weakened to mere existence of a nonnegative-mass end

ordered sign-pattern encoded by the index $\Lambda$

nonnegative conclusion strengthened to rigidity/strict positivity without the theorem's equality analysis


\label{thm:main-strong-cover}
    Let $\varepsilon > 0$. There exists a constant $C = C(\varepsilon)>0$ such that if $G$ is an $(n,d,\lambda)$-graph with $d/\lambda \geq C$, then $ST(G) \leq (1+\varepsilon)n\log n$.

\begin{proposition}\label{prop:cover-time}
    For every $\varepsilon >0$, there exists $C = C(\eps)$ such that if $G$ be an $(n,d,\lambda)$-graph with $d/\lambda \geq C$, then $T(G) \leq (1+\eps) n\log n$.
\end{proposition}

\begin{theorem}\label{thm:main-strong-cover}
    Let $\eps > 0$. There exists a constant $C = C(\eps)>0$ such that if $G$ is an $(n,d,\lambda)$-graph with $d/\lambda \geq C$, then $ST(G) \leq (1+\eps)n\log n$.
\end{theorem}

Let $G$ be a simple undirected connected graph. A \emph{simple random walk} on $G$ is a stochastic process where, starting from a given vertex, each step consists of moving to a neighbor chosen uniformly at random. The set of edges traversed by this walk is called the \emph{trace}. Random walks are fundamental objects in combinatorics, probability theory, and theoretical computer science. For a comprehensive treatment, we refer the reader to the textbook by Levin and Peres~\cite{Levin-Peres-book} or the survey by Lov\'{a}sz~\cite{Lovasz-survey}.

In this paper, we study the graph formed by the trace of a simple random walk on a finite graph. This line of research was initiated by Frieze, Krivelevich, Michaeli, and Peled in~\cite{Frieze-et-al}, who focused on the case where the base graph is random. Motivated by these results, in~\cite{Frieze-et-al}, they asked to extend the inquiry to deterministic graphs that exhibit random-like properties, known as pseudorandom graphs. A prominent class of pseudorandom graphs is that of spectral expanders, or $(n, d, \lambda)$-graphs. An $(n, d, \lambda)$-graph is defined as a $d$-regular graph on $n$ vertices where the second largest eigenvalue in absolute value is at most $\lambda$. The expander mixing lemma shows that $\lambda$ governs the edge distribution: a smaller $\lambda$ implies that the edge distribution of $G$ closely resembles that of the random graph $G(n, d/n)$. For a detailed introduction, we refer the reader to the survey by Krivelevich and Sudakov~\cite{Krivelevich-Sudakov-survey}.

We focus our attention on the case where $G$ is an $(n,d,\lambda)$-graph. Let $\Gamma$ denote the subgraph on the vertex set of $G$ induced by the trace of the random walk. A fundamental question concerns the structural properties of $\Gamma$, particularly its connectivity. This inquiry is intrinsically linked to the \emph{cover time}, $T(G)$, defined as the expected number of steps required to visit every vertex, maximized over all possible starting vertices. Equivalently, $T(G)$ corresponds to the expectation of the hitting time for the property that $\Gamma$ becomes connected. The cover time is a classical parameter in the study of random walks. A seminal result by Feige~\cite{Feige-lower} establishes a universal lower bound of $(1-o(1))n\log n$ for any connected graph on $n$ vertices. For Erd\H{o}s-R\'{e}nyi graph, random regular graph and more generally random graph with a fixed degree sequence, asymptotically sharp bounds have been proven~\cite{Adullah-Copper-Frieze-degree-sequence,Cooper-Frieze-sparse-random, Cooper-Frieze-random-regular,Jonasson, Cooper-Frieze-Lubetzky}. In particular, Frieze, Krivelevich, Michaeli, and Peled~\cite{Frieze-et-al} demonstrated that for random graphs $G(n,p)$ and $\varepsilon>0$, there exists a constant $C_{\varepsilon}$ such that if $p \ge C_{\varepsilon}\log n/n$, the trace $\Gamma$ generated by a random walk of length $(1+\varepsilon)n\log n$ is not merely connected, but Hamiltonian with high probability.

Given that $(n,d,\lambda)$-graphs exhibit quasi-random behavior analogous to the random graph $G(n, d/n)$, it is natural to inquire whether the structural properties of the random walk trace keep in this setting. Specifically, we ask whether a random walk of length $(1+\varepsilon)n\log n$ yields a Hamiltonian trace on an $(n,d,\lambda)$-graph. This question was raised by Frieze, Krivelevich, Michaeli, and Peled~\cite{Frieze-et-al}.
Recall that the connectivity threshold for $G(n,p)$ is $(1+ o(1))\log n/n$ where as for the trace to be Hamiltonian, one needs $C_\varepsilon\log /n$ where $C_\varepsilon$ is a big constant depending on $\varepsilon$. For $(n,d,\lambda)$-graph, the connectivity holds when $d/\lambda > 1$. Thus, analogously, one would expect the answer holds when $d/\lambda > C_\varepsilon$ for some big constant $C_\varepsilon$.

\subsection{Our results}
By the expander mixing lemma, $(n,d,\lambda)$-graphs represent a class of graphs with expansion properties. The asymptotic behavior of random walks in this setting was investigated in the foundational work of Broder and Karlin~\cite{Broder-Kralin}, who established that for any $d$-regular expander $G$, the cover time satisfies $T(G) = \Theta(n\log n)$ \footnote{The notion of expander in~\cite{Broder-Kralin} is different, but it applies to the $(n,d,\lambda)$ setting.}. Our first contribution is a refinement of this bound for $(n,d,\lambda)$-graphs. We demonstrate that, provided the spectral ratio is sufficiently large, the cover time is asymptotically optimal: $T(G) = (1+o(1))n\log n$.
\begin{proposition}\label{prop:cover-time}
 For every $\varepsilon >0$, there exists $C = C(\varepsilon)$ such that if $G$ be an $(n,d,\lambda)$-graph with $d/\lambda \geq C$, then $T(G) \leq (1+\varepsilon) n\log n$.
\end{proposition}

We now introduce a strengthrening of the cover time, termed the \emph{strong cover time}. Let $ST(G)$ denote the minimum integer $t$ such that a simple random walk of length $t$ visits every vertex of $G$ with high probability (abbrviated as \textbf{whp} throughout the paper), uniformly over all starting vertices. Note that by definition, $ST(G)$ provides a high-probability guarantee, whereas $T(G)$ captures only the expectation.
Deterministically $ST(G)\geq T(G)$.
In the context of random graphs $G(n,p)$, Jonasson~\cite{Jonasson} established that $ST(G)= (1+o(1))n\log n$ provided that $p = \omega(\log^3n/n)$. This result was subsequently refined by Frieze, Krivelevich, Michaeli, and Peled~\cite{Frieze-et-al}, who extended the bound to the asymptotically optimal $p = \omega(\log n/n)$. We consider the problem analogous to the $(n,d,\lambda)$-graphs.

\begin{theorem}\label{thm:main}
 Let $\varepsilon > 0$. There exists a constant $C = C(\eps) >0$ such that the following holds: Let $G$ be an $(n,d,\lambda)$-graph with $d/\lambda \geq C$ and $L = (1+\eps) n\log n$. \textbf{whp} for every vertex $v\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.
\end{theorem}

It is well known that a random $d$-regular graph for any $d\geq 3$ is \textbf{whp} an $(n,d,\lambda)$-graph for $\lambda = O(\sqrt{d})$~\cite{Friedman}. We thus obtain the following corollary.
\begin{corollary}\label{cor:random-d-regular}
 Let $\varepsilon > 0$. There exists $d_0 = d_0(\eps)$ that the following holds: Let $G$ be a random $d$-regular graph with $d\geq d_0$ and let $L = (1+\eps) n\log n$. \textbf{whp} $G$ satisfies that \textbf{whp} for every vertex $v\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.
\end{corollary}
We remark that both results are asymptotically optimal due to the general lower bound on the cover time. Our proof strategy is as follows.
First, since the second largest eigenvalue of the adjacency matrix of an $(n,d,\lambda)$-graph is bounded away from $d$, we can control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.
We show that after $(1+\varepsilon)n\log n$ steps, each vertex is visited $\Theta(\log n)$ times with high probability. Then, we use these visit counts to establish the expansion properties of the trace graph $\Gamma$, which ensures the existence of a Hamiltonian cycle.

\subsection{Number of visits}
For a random walk on a graph $G$ of length $L$ and a vertex $v\in V(G)$, let $\gamma(v)$ denote the number of times the random walk visits $v$.
The goal of this section is to prove the following:
\begin{lemma}\label{lem:number-of-vists}
 Let $\varepsilon > 0$. There exists constants $C = C(\eps) >0$ and $\rho = \rho(\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\lambda)$-graph with $d/\lambda \geq C$ and $L = (1+\eps) n\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \textbf{whp} $\gamma(v)\geq \rho \log n$ for every vertex $v\in V(G)$.
\end{lemma}
We begin by establishing an auxiliary lemma, giving a lower bound on the probability a random walk starting from $u$ reaches $v$ in $n/\sqrt{C}$ steps:
\begin{lemma}\label{lem:lower-bound-return}
 Let $\varepsilon > 0$. There exists a constant $C = C(\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\lambda)$-graph with $d/\lambda \geq C$ and $T = n/\sqrt{C}$. Then, for any pair of distinct vertices $u,v\in V(G)$, the probability a simple random walk $X$ starting from $u$ reaches $v$ at least once within $T$ steps is at least $(1-0.1\eps)/\sqrt{C}$.
\end{lemma}
\begin{proof}
 Let $N$ denote the number of visits to $v$ in the first $T$ steps and $I_t$ the indicator random variable for $X_t = v$ so that $N = \sum_{t = 1}^T I_i$. By linearity of expectation, we have
 \[
 \E[N] = \sum_{t=1}^T \Pr(X_t = v|X_0 = u) = \sum_{t=1}^T (P^t)_{u,v}\,.
 \]
 Recall that the transition matrix $P = D^{-1}A$ and let $d = \lambda_1\geq \dots\geq \lambda_n$ are eigenvalues of $A$. Define $S = D^{1/2}PD^{-1/2} = D^{-1/2}AD^{-1/2}$. Since $S$ is a symmetric real square matrix, its spectral decomposition can be written as $S = \sum_{i=1}^n\mu_i\phi_i \phi_i^T$, where $\mu_i = \lambda_i/d\in [-1,1]$ are eigenvalues of $S$ and $\{\phi_i\}_{i=1}^n$ forms an orthonormal basis.
 Observe that the stationary distribution of $X$ is $\pi = (1/n,\dots, 1/n)$, $\mu_1 = 1$ and $\phi_1(v) = (1/\sqrt{n},\dots, 1/\sqrt{n})$.

We have the following immediate corollary for the stationary walk:
\begin{corollary}\label{cor:stationary-walk}
 Let $\varepsilon > 0$. There exists a constant $C = C(\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\lambda)$-graph with $d/\lambda \geq C$ and $T = n/\sqrt{C}$. Then, for any vertex $v\in V(G)$, the probability a stationary walk visits $v$ at least once within $T$ steps is at least $(1-0.1\eps)/\sqrt{C}$.
\end{corollary}
\begin{proof}
 Recall that the stationary distribution for a $d$-regular graph is the uniform distribution over all vertices $(1/n,\dots, 1/n)$. Also, the stationary walk $Y$ and the simple random walk $X$ have the same distribution conditionally on having the same starting vertex.
 Since~\Cref{lem:lower-bound-return} holds for every starting vertex $u\in V(G)$, the conclusion follows by integrating over the starting vertex.
\end{proof}

\subsection{C-expander}
The goal of this section is to prove the following:
\begin{lemma}\label{lem:C-expander}
 Let $\varepsilon > 0$. There exists a constant $C = C(\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\lambda)$-graph with $d/\lambda \geq C$ and $L = (1+\eps) n\log n$.
 Let $\Gamma$ be the trace graph obtained by a random walk of length $L$ starting from any vertex. Then,
 \textbf{whp} $\Gamma$ is a $C'$-expander for $C' = \log \log C$.
\end{lemma}
\begin{proof}
 To prove a graph is $C$-expander, we need to verify the expansion and joinedness properties. By~\Cref{lem:number-of-vists}, there exists a constant $\rho > 0$ such that $\gamma(v) \geq \rho \log n$ for all $v\in V(G)$.

\section{Concluding remarks}\label{sec:concluding-remarks}
One of the key tools we applied was~\Cref{thm:Hamiltonicity}, which requires only $C$-expanders. One may wonder if~\Cref{thm:main} is true with $(n,d,\lambda)$-graphs replaced by $C$-expanders. This is unfortunately (very) incorrect:
\begin{proposition}
 For any $C \leq 1.1n/\log n$, there exists a $C$-expander for which the cover time is $a n \log n$ for some $a > 1$.
\end{proposition}
\begin{proof}[Proof sketch]
 Consider a complete graph on $n-2$ vertices together with two special vertices $u,v$, each connecting to $C$ vertices of the complete graph and their neighborhoods are disjoint. Clearly, the graph is a $C$-expander. However, note that the expected number of steps needed to go from $u$ to $v$ is at least $1 + (n-1)/C)\cdot (n-1)> n\log n$. Indeed, the $1$ comes from going from $u$ to the clique. Then, in each step, it will have at most a $C/(n-1)$ chance of landing in $N(v)$ and thus in expectation $(n-1)/C$ steps are needed to land in $N(v)$. Starting from a vertex in $N(v)$, each time there is a $1/(n-1)$ chance of going to $v$ so in expectation it takes $n-1$ such trials. Clearly, the expected number of steps taken to go from $u$ to $v$ is a lower bound on the cover time, which finishes the proof.
\end{proof}
One natural strengthening of~\Cref{thm:main} would be to consider following hitting time problem: Let $\tau_{HC}(G)$ be the minimum $t\in \mathbb N$ for which the trace graph $\Gamma$ becomes Hamiltonian and $\tau_{1}$ the minimum $t\in \mathbb N$ such that $\Gamma$ has minimum degree $1$.
For $(n,d,\lambda)$-graphs with $d/\lambda \geq C$ or a random $d$-regular graph for $d$ sufficiently large, is it true that \textbf{whp} $\tau_{HC} = \tau_1 + 1$? Note that $\tau_{HC}\geq \tau_1 + 1$ is necessary since a Hamilton cycle uses two incident edges from each vertex.
For complete graph the corresponding hitting time result is known~\cite{Frieze-et-al} .

Again this is asymptotically optimal by the general lower bound of the cover time. In fact, we obtain~\Cref{thm:main-strong-cover} as a direct consequence of a stronger result.

The connection between spectral expansion and the Hamiltonicity in pseudorandom graphs is well-established as in~\cite{Kriveham,glockham,ferber2024hamiltonicity,chen2025robustness}. A recent breakthrough by Draganić, Montgomery, Munhá Correia, Pokrovskiy and Sudakov~\cite{Draganic-et-al} demonstrates that any $(n,d,\lambda)$-graph admits a Hamiltonian cycle, if the spectral ratio $d/\lambda\ge C$ for a sufficiently large constant $C$. Drawing a parallel to the trace of random walk landscape, Frieze, Krivelevich, Michaeli, and Peled~\cite{Frieze-et-al} raised the question of whether the trace of a random walk on $(n,d,\lambda)$-graphs and random regular graphs exhibits similar Hamiltonicity properties.
We answer these questions through the following two main results.

\begin{theorem}\label{thm:main}
    Let $\varepsilon > 0$. There exists a constant $C = C(\varepsilon) >0$ such that the following holds: Let $G$ be an $(n,d,\lambda)$-graph with $d/\lambda \geq C$ and $L = (1+\varepsilon) n\log n$. \textbf{whp} for every vertex $v\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.
\end{theorem}

We remark that implicitly in~\Cref{prop:cover-time},~\Cref{thm:main-strong-cover} and~\Cref{thm:main}, $d$ is sufficiently large. This follows from the fact that $\lambda = \Omega(d)$ (see~\cite{Krivelevich-Sudakov-survey}), which means $1/C \geq \lambda/d\geq \Omega(1/\sqrt{d})$ and therefore $d$ is sufficiently large as long as $C$ is.

It is well known that a random $d$-regular graph for any $d\geq 3$ is \textbf{whp} an $(n,d,\lambda)$-graph for $\lambda = O(\sqrt{d})$~\cite{Friedman}. We thus obtain the following corollary.
\begin{corollary}\label{cor:random-d-regular}
     Let $\varepsilon > 0$. There exists $d_0 = d_0(\varepsilon)$ that the following holds: Let $G$ be a random $d$-regular graph with $d\geq d_0$ and let $L = (1+\varepsilon) n\log n$. \textbf{whp} $G$ satisfies that \textbf{whp} for every vertex $v\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.
\end{corollary}
We remark that both results are asymptotically optimal due to the general lower bound on the cover time. Our proof strategy is as follows.
First, since the second largest eigenvalue of the adjacency matrix of an $(n,d,\lambda)$-graph is bounded away from $d$, we can control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.
We show that after $(1+\varepsilon)n\log n$ steps, each vertex is visited $\Theta(\log n)$ times with high probability. Then, we use these visit counts to establish the expansion properties of the trace graph $\Gamma$, which ensures the existence of a Hamiltonian cycle.

Central to our proof of \Cref{thm:main} is the lemma that the trace of the random walk visits every vertex with logarithmic frequency. Specifically, \Cref{lem:number-of-vists} establishes that there exists a constant $\rho = \rho(\varepsilon) > 0$ such that with high probability, every vertex is visited at least $\rho \log n$ times. This phenomenon is intimately connected to the concept of \emph{blanket time} in the probability theory, a parameter introduced by Winkler and Zuckerman~\cite{Winkler-Zuckerman}. Let $\pi$ denote the stationary distribution, where $\pi_v = \deg(v)/(2|E(G)|)$. For a parameter $\delta \in (0,1)$, the $\delta$-blanket time, denoted $t_{bl}(G,\delta)$, is defined as the minimum time $t$ required for the walk to visit every vertex $v$ at least $\delta t \pi_v$ times. While it is immediate that $t_{bl}(G,\delta) \ge T(G)$, Winkler and Zuckerman conjectured that the blanket time is, in fact, within a constant factor of the cover time; that is, $t_{bl}(G,\delta) \le C_\delta T(G)$ for any $0<\delta < 1$ and any graph $G$.
Kahn, Kim, Lov{\'a}sz and Vu~\cite{Kahn-Kim-Lovasz-Vu} showed that $t_{bl}(G,\delta)= O(T(G)(\log\log n)^2)$ before the conjecture was affirmatively settled in a celebrated work of Ding, Lee, and Peres~\cite{Ding-Lee-Peres}.
Our results imply that there exists a $\delta = \delta(\varepsilon)\in (0,1)$ for which $t_{bl}(G,\delta)\leq (1+\varepsilon)n\log n$, which is the same as the cover time up to a $1+o(1)$ factor. In general, it is unclear for which graphs the same conclusion hold. In the concluding remarks we discuss and propose further questions along this direction.

\subsection{Paper organization}
\Cref{sec:preliminaries} contains the background of random walks and tools we use.~\Cref{sec:proof} contains the proofs for all results and~\Cref{sec:concluding-remarks} some remarks and open problems.

\subsection*{Acknowledgement}
Part of this work was initiated during the visit of second author to Shanghai Center for Mathematical Sciences (SCMS) and Fudan University. The authors would like to thank Prof.\ Hehui Wu and his group memebers for their support and hospitality.

They state that \Cref{thm:main-strong-cover} is obtained “as a direct consequence of a stronger result,” namely \Cref{thm:main}. The proof strategy outlined is:
\begin{itemize}
\item Use that “the second largest eigenvalue of the adjacency matrix of an $(n,d,\lambda)$-graph is bounded away from $d$” to “control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.”
\item Show that “after $(1+\varepsilon)n\log n$ steps, each vertex is visited $\Theta(\log n)$ times with high probability.” (Central is the lemma that “with high probability, every vertex is visited at least $\rho\log n$ times,” i.e. \Cref{lem:number-of-vists}.)
\item “Use these visit counts to establish the expansion properties of the trace graph $\Gamma$, which ensures the existence of a Hamiltonian cycle.” Since \Cref{thm:main-strong-cover} follows directly from this stronger Hamiltonicity statement, it follows as a corollary.
\end{itemize}
They further connect the logarithmic visit-frequency lemma to “blanket time,” noting their results imply the existence of $\delta=\delta(\varepsilon)\in(0,1)$ with $t_{bl}(G,\delta)\le (1+\varepsilon)n\log n$.

They state that \Cref{thm:main-strong-cover} is obtained “as a direct consequence of a stronger result,” namely the stronger Hamiltonicity statement.

The proof strategy outlined is:
\begin{itemize}
\item Use that “the second largest eigenvalue of the adjacency matrix of an $(n,d,\lambda)$-graph is bounded away from $d$” to “control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.”
\item Show that “after $(1+\varepsilon)n\log n$ steps, each vertex is visited $\Theta(\log n)$ times with high probability.” Central is the following lemma.

\begin{lemma}\label{lem:number-of-vists}
    Let $\varepsilon > 0$. There exists constants $C = C(\eps) >0$ and $\rho = \rho(\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\lambda)$-graph with $d/\lambda \geq C$ and $L = (1+\eps) n\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \textbf{whp} $\gamma(v)\geq \rho \log n$ for every vertex $v\in V(G)$.
\end{lemma}

\item “Use these visit counts to establish the expansion properties of the trace graph $\Gamma$, which ensures the existence of a Hamiltonian cycle.” In establishing the main theorem, this yields Hamiltonicity of the trace, and hence the stated bound on $ST(G)$ follows as a corollary.
\end{itemize}
They further connect the logarithmic visit-frequency lemma to “blanket time,” noting their results imply the existence of $\delta=\delta(\varepsilon)\in(0,1)$ with $t_{bl}(G,\delta)\le (1+\varepsilon)n\log n$.

Question: Let \(\varepsilon>0\). An \((n,d,\lambda)\)-graph is a \(d\)-regular graph on \(n\) vertices whose second-largest eigenvalue in absolute value is at most \(\lambda\). Let \(ST(G)\) denote the strong cover time of \(G\), i.e. the least integer \(t\) such that a simple random walk of length \(t\), started from any vertex of \(G\), covers all vertices with high probability. Which conclusion is valid under the assumption that \(G\) is an \((n,d,\lambda)\)-graph with \(d/\lambda\ge C\) for a suitable constant \(C=C(\varepsilon)>0\)?

For every \(\varepsilon>0\), there exists a constant \(C=C(\varepsilon)>0\) such that if \(G\) is an \((n,d,\lambda)\)-graph with \(d/\lambda\ge C\), then \(ST(G)\le (1+\varepsilon)n\log n\).

Choices:
B. For every \(\varepsilon>0\), there exists a constant \(C=C(\varepsilon)>0\) such that if \(G\) is an \((n,d,\lambda)\)-graph with \(d/\lambda\ge C\), then for every starting vertex, a simple random walk of length exactly \((1+\varepsilon)n\log n\) visits each vertex at least once with high probability.
C. For every \(\varepsilon>0\), there exists a constant \(C=C(\varepsilon)>0\) such that if \(G\) is an \((n,d,\lambda)\)-graph with \(d/\lambda\ge C\), then the ordinary cover time of \(G\) is at most \((1+\varepsilon)n\log n\).
D. There exists a universal constant \(C>0\) such that for every \(\varepsilon>0\), if \(G\) is an \((n,d,\lambda)\)-graph with \(d/\lambda\ge C\), then \(ST(G)\le (1+\varepsilon)n\log n\).
E. For every \(\varepsilon>0\), there exists a constant \(C=C(\varepsilon)>0\) such that if \(G\) is an \((n,d,\lambda)\)-graph with \(d/\lambda\ge C\), then after \((1+\varepsilon)n\log n\) steps, a simple random walk started from any vertex visits every vertex at least \(\rho\log n\) times with high probability for some constant \(\rho=\rho(\varepsilon)>0\).


[\textsc{Weighted Khintchine for manifolds}]\label{thm: dream0}
Let $\mathcal{M}$ be any nondegenerate submanifold of $\mathbb{R}^n$ and $\psi_1,\dots,\psi_n:\mathbb{R}_+\to(0,1)$ be non-increasing. Then almost all points on $\mathcal{M}$ are $(\psi_1,\dots,\psi_n)$-approximable if the series
\begin{equation}\label{mainsum}
\sum_{q=1}^\infty\psi_1(q)\cdots \psi_n(q)
\end{equation}
diverges, and are not $(\psi_1,\dots,\psi_n)$-approximable if \eqref{mainsum} converges.

\begin{theorem}[\textsc{Weighted Khintchine for manifolds}]\label{thm: dream0}
Let $\cM$ be any nondegenerate submanifold of $\R^n$ and $\psi_1,\dots,\psi_n:\Rp\to(0,1)$ be non-increasing. Then almost all points on $\cM$ are $(\psi_1,\dots,\psi_n)$-approximable if the series
\begin{equation}\label{mainsum}
\sum_{q=1}^\infty\psi_1(q)\cdots \psi_n(q)
\end{equation}
diverges, and are not $(\psi_1,\dots,\psi_n)$-approximable if \eqref{mainsum} converges.
\end{theorem}

\begin{equation}\label{eq: bound on derivatives}
   \max_{k}\max_{i,j} \sup_{\bx\in \bU}\vert\partial_{ij}f_{k}(\bx)\vert\leq M.
\end{equation}

\subsection{Quantitative Nondivergence}\label{new BKM}

Our proofs will require an effective measure estimate for the following set
\begin{align}
\mathfrak{S}_{F}&(\delta,\bK,\bT)=\nonumber\\[1ex]
&=\left\{\bx\in \bU:\exists\;(a_0,\ba)\in\Z\times\Z^n_{\neq\bf0}\;\;\text{such that }\left.
\begin{array}{l}
|a_0+F(\bx)\ba^\top|<\delta\\[1ex]
\vert \partial_iF(\bx)\ba^\top\vert<K_i\;\;(1\le i\le d) \\[1ex]
\vert a_k\vert<T_k ~~(1\leq k\leq n)
\end{array}
\right.\right\},\label{eq: new set}
\end{align}
where $\ba=(a_1,\dots,a_n)$, $\bT=(T_1,\dots,T_n)$, $\bK=(K_1,\dots,K_d)$, and $\delta,K_1,\cdots,K_d,T_1,\cdots, T_n$ are positive real parameters.

The following result represents a generalisation of \cite[Theorem 1.4]{BKM}
and appears in \cite[Theorem 3.1]{BD} when $K_1=\dots=K_d$. Its proof in the general case follows the same lines as that of \cite[Theorem 3.1]{BD} with minor modifications which will be explained below.

\begin{theorem}\label{BKM_new2}
Let $F:\bU\to\R^n$ be as defined in \eqref{eq: F}, $l$-nondegenerate at $\bx_0$.
Then there exists a ball $B_0\subset\bU$ centred at $\bx_0$ and a constant $E>0$ depending on $B_0$ and $F$ such that for any choice of $\delta,K_1,\cdots,K_d,T_1,\cdots, T_n$ satisfying
\begin{equation}\label{eqn5.2}
0<\delta\le 1,\qquad T_1,\dots,T_n\ge1,\qquad K_1,\cdots,K_d>0 \quad\text{and}\quad \delta^{n}<K \frac{T_1\cdots T_n}{\max_i T_i},
\end{equation}

One classical theme in the theory of Diophantine approximation is to quantify how well rational points
$$
\mathbf{p}/q=(p_1/q,\dots,p_n/q)\,,
$$
with $q\in\mathbb{N}$ and $\mathbf{p}\in\mathbb{Z}^n$, approximate real points $\mathbf{x}=(x_1,\dots,x_n)\in\mathbb{R}^n$.
In the standard setting, the error of approximations is measured using a fixed norm on $\mathbb{R}^n$, usually the sup-norm
$$
\|\mathbf{x}\|=\max_{1\le i\le n}|x_i|\,.
$$
Thus one seeks the same rate of approximation in each coordinate, imposing identical bounds on the quantities $|x_i-p_i/q|$.

In this paper we consider two more general frameworks: \emph{weighted} and \emph{multiplicative}, both of which can be seen as important special cases of the more general theory of Diophantine approximation with star bodies \cite{MR2264601}. In the weighted case, different coordinates may be approximated at different rates; in the multiplicative case, one specifies a bound on the product
$\prod_{i=1}^n |x_i - p_i/q|$.
We now introduce the relevant notation and terminology.

Let $\mathbb{R}_+$ denote the set of positive real numbers, and let $\psi,\psi_1,\dots,\psi_n:\mathbb{R}_+\to(0,1)$ be nonincreasing  functions, referred to as {\em approximation functions}.
Define $\mathcal{S}_n(\psi_1,\dots, \psi_n)$ as the set of $\mathbf{x}\in\mathbb{R}^n$ such that the system
\begin{equation}\label{eq1.1}
\vert qx_i-p_i\vert<\psi_i(q)\qquad (1\leq i\leq n)
\end{equation}
holds for infinitely many $(q,p_1,\dots,p_n)\in\mathbb{N}\times\mathbb{Z}^n$. When $\psi_1=\dots=\psi_n=\psi$ this set will be denoted by $\cS_n(\psi)$. A point $\mathbf{x}\in\cS_n(\psi_1,\dots,\psi_n)$ will be called {\em $(\psi_1,\dots,\psi_n)$--approximable} ({\em $\psi$-approximable} when $\psi_1=\dots=\psi_n=\psi$).

In the multiplicative case,
define $\cSM_n(\psi)$ as  the set of $\mathbf{x}\in\mathbb{R}^n$ such that
$$  \prod_{i=1}^n |qx_i-p_i| \ < \ \psi(q)
$$
holds for infinitely many $(q,p_1,\dots,p_n)\in\mathbb{N}\times\mathbb{Z}^n$. The points in $\cSM_n(\psi)$ will be called
{\em multiplicatively
$\psi$--approximable}.
It is readily verified that
\begin{equation}\label{s:004}
  \cS_n(\psi_1,\dots,\psi_n)\subset\cSM_n(\psi)\quad\text{if}\quad\psi\ge\psi_1\cdots\psi_n\,.
\end{equation}

\noindent By Minkowski's linear forms theorem,
\begin{equation}\label{sv:004}
\cS_n(\psi_1,\dots,\psi_n)=\mathbb{R}^n \hspace{9mm}  {\rm  if } \hspace{9mm}
q\psi_1(q)\cdots\psi_n(q)\ge 1 \quad\text{infinitely often}\,.
\end{equation}
This together with \eqref{s:004} implies that
\begin{equation}\label{sv:005}
\cSM_n(\psi)=\mathbb{R}^n  \hspace{9mm}  {\rm  if } \hspace{9mm} q\psi(q)\ge 1 \quad\text{infinitely often} .
\end{equation}
Littlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \eqref{sv:005} can be weakened to
$$\psi(q)\ge \varepsilon q^{-1}\qquad\text{for any $\varepsilon>0$},
$$
see \cite{EKL} and \cite{PVActa}. However, in the weighted case,
such a weakening is generally impossible for the existence
of badly approximable points \cite{MR2581371, MR2231044}, e.g. when
$$
\psi_i(q)=q^{-\tau_i}\,,\qquad \tau_i\ge0,\quad\tau_1+\dots+\tau_n=1\,.
$$
Here $\mathbf{x}\in\mathbb{R}^n$ is {\em $(\psi_1,\dots,\psi_n)$-badly approximable} if $\mathbf{x}\not\in\cS_n(c\psi_1,\dots,c\psi_n)$ for some $c>0$. Nevertheless, the set of $(\psi_1,\dots,\psi_n)$-badly approximable points is well known to have zero measure \cite{BadNull}.

In this paper we will be concerned with the measure of the sets $\cS_n(\psi_1,\dots,\psi_n)$ and $\cSM_n(\psi)$
restricted to submanifolds of $\mathbb{R}^n$ when $\psi_1,\dots,\psi_n,\psi$ are generic non-increasing functions.
We begin by recalling the corresponding results for $\mathbb{R}^n$. Let $\mu_n$ denote Lebesgue measure on $\mathbb{R}^n$.

\smallskip

\begin{thkh}
\label{gs} Let   $\psi_1,\dots,\psi_n$ be non-increasing functions. Then
$$\mu_{n}\left(\cS_n(\psi_1,\dots,\psi_n)\right) =\left\{\begin{array}{ll} \mbox{\rm
Z{\scriptsize ERO}} & {\rm if} \;\;\; \sum \;  \psi_1(q)\cdots \psi_n(q)
\;\; <\infty\,,\\ &
\\[-1ex] \mbox{\rm F{\scriptsize ULL}} & {\rm if} \;\;\; \sum  \;  \psi_1(q)\cdots \psi_n(q) \;\;
 =\infty\,. \; \;
\end{array}\right.$$
\end{thkh}

\smallskip

\begin{thga}
\label{gm} Let   $\psi$ be a non-increasing function. Then
$$\mu_{n}\left(\cSM_n(\psi)\right) =\left\{\begin{array}{ll} \mbox{\rm
Z{\scriptsize ERO}} & {\rm if} \;\;\; \sum \;  \psi(q) \ (\log
q)^{n-1} \;\; < \ \infty\,,\\ &
\\[-1ex] \mbox{\rm F{\scriptsize ULL}} & {\rm if} \;\;\; \sum  \;  \psi(q) \ (\log
q)^{n-1}  \;\;
 = \ \infty\,. \; \;
\end{array}\right.$$
\end{thga}

\noindent Here `\mbox{\rm F{\scriptsize ULL}}' means that the complement has  measure zero. Theorem~K was originally established by Khintchine
\cite{Kh} when $\psi_1=\dots=\psi_n$, and both results as stated above were obtained by Gallagher in \cite{gal}.

Our main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\mathbb{R}^n$ as defined in \cite{KM}.

\smallskip

Let $\mathbb{R}_+$ denote the set of positive real numbers, and let $\psi,\psi_1,\dots,\psi_n:\mathbb{R}_+\to(0,1)$ be nonincreasing functions, referred to as {\em approximation functions}.
Define $\mathcal{S}_n(\psi_1,\dots, \psi_n)$ as the set of $\mathbf{x}\in\mathbb{R}^n$ such that the system
\begin{equation}\label{eq1.1}
\vert qx_i-p_i\vert<\psi_i(q)\qquad (1\leq i\leq n)
\end{equation}
holds for infinitely many $(q,p_1,\dots,p_n)\in\mathbb{N}\times\mathbb{Z}^n$. When $\psi_1=\dots=\psi_n=\psi$ this set will be denoted by $\cS_n(\psi)$. A point $\mathbf{x}\in\cS_n(\psi_1,\dots,\psi_n)$ will be called {\em $(\psi_1,\dots,\psi_n)$--approximable} ({\em $\psi$-approximable} when $\psi_1=\dots=\psi_n=\psi$).

In the multiplicative case,
define $\cSM_n(\psi)$ as the set of $\mathbf{x}\in\mathbb{R}^n$ such that
$$ \prod_{i=1}^n |qx_i-p_i| \ < \ \psi(q)
$$
holds for infinitely many $(q,p_1,\dots,p_n)\in\mathbb{N}\times\mathbb{Z}^n$. The points in $\cSM_n(\psi)$ will be called
{\em multiplicatively
$\psi$--approximable}.
It is readily verified that
\begin{equation}\label{s:004}
 \cS_n(\psi_1,\dots,\psi_n)\subset\cSM_n(\psi)\quad\text{if}\quad\psi\ge\psi_1\cdots\psi_n\,.
\end{equation}

\noindent By Minkowski's linear forms theorem,
\begin{equation}\label{sv:004}
\cS_n(\psi_1,\dots,\psi_n)=\mathbb{R}^n \hspace{9mm} {\rm if } \hspace{9mm}
q\psi_1(q)\cdots\psi_n(q)\ge 1 \quad\text{infinitely often}\,.
\end{equation}
This together with \eqref{s:004} implies that
\begin{equation}\label{sv:005}
\cSM_n(\psi)=\mathbb{R}^n \hspace{9mm} {\rm if } \hspace{9mm} q\psi(q)\ge 1 \quad\text{infinitely often} .
\end{equation}
Littlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \eqref{sv:005} can be weakened to
$$\psi(q)\ge \varepsilon q^{-1}\qquad\text{for any $\varepsilon>0$},
$$
see \cite{EKL} and \cite{PVActa}. However, in the weighted case,
such a weakening is generally impossible for the existence
of badly approximable points \cite{MR2581371, MR2231044}, e.g. when
$$
\psi_i(q)=q^{-\tau_i}\,,\qquad \tau_i\ge0,\quad\tau_1+\dots+\tau_n=1\,.
$$
Here $\mathbf{x}\in\mathbb{R}^n$ is {\em $(\psi_1,\dots,\psi_n)$-badly approximable} if $\mathbf{x}\not\in\cS_n(c\psi_1,\dots,c\psi_n)$ for some $c>0$. Nevertheless, the set of $(\psi_1,\dots,\psi_n)$-badly approximable points is well known to have zero measure \cite{BadNull}.

\begin{thkh}
\label{gs} Let $\psi_1,\dots,\psi_n$ be non-increasing functions. Then
$$\mu_{n}\left(\cS_n(\psi_1,\dots,\psi_n)\right) =\left\{\begin{array}{ll} \mbox{\rm
Z{\scriptsize ERO}} & {\rm if} \;\;\; \sum \; \psi_1(q)\cdots \psi_n(q)
\;\; <\infty\,,\\ &
\\[-1ex] \mbox{\rm F{\scriptsize ULL}} & {\rm if} \;\;\; \sum \; \psi_1(q)\cdots \psi_n(q) \;\;
 =\infty\,. \; \;
\end{array}\right.$$
\end{thkh}

Our main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\mathbb{R}^n$ as defined in \cite{KM}.

\smallskip

\begin{equation}\label{mainsum}
\sum_{q=1}^\infty\psi_1(q)\cdots \psi_n(q)
\end{equation}

\begin{equation}\label{s:004}
  \cS_n(\psi_1,\dots,\psi_n)\subset\cSM_n(\psi)\quad\text{if}\quad\psi\ge\psi_1\cdots\psi_n\,.
\end{equation}

\begin{equation}\label{sv:005}
\cSM_n(\psi)=\R^n  \hspace{9mm}  {\rm  if } \hspace{9mm} q\psi(q)\ge 1 \quad\text{infinitely often} .
\end{equation}

\smallskip

\medskip

\begin{theorem}[\textsc{Multiplicative convergence for manifolds}]\label{thm:mult0}Let $\cM$ be any nondegenerate submanifold of $\R^n$ and $\psi:\Rp\to(0,1)$ be non-increasing. Then almost all points on $\cM$ are not multiplicatively
$\psi$-approximable if
$$
\sum_{q=1}^\infty \psi(q) (\log q)^{n-1}<\infty.
$$
\end{theorem}

\begin{theorem}\label{thm: dream}
Let $F:\bU\to\R^n$ be nondegenerate, $\bU$ be an open subset of $\R^d$, $\psi_1,\dots,\psi_n:\Rp\to(0,1)$ be non-increasing. Then \begin{equation}
 \mu_d\left(F^{-1}\mathcal{S}_n(\psi_1,\dots, \psi_n)\right)=\left\{\begin{aligned}
 & 0 &\text{ if } &\textstyle\sum\psi_1(q)\cdots \psi_n(q)<\infty\,,\\[1ex]
 & \mu_d(\bU) &\text{ if } &\textstyle\sum \psi_1(q)\cdots \psi_n(q)=\infty\,.
 \end{aligned}\right.
 \end{equation}
\end{theorem}

With this proposition at hand, justifying \eqref{eq: prod is greater than} becomes simple. We take $\Phi(q)=q^{-1-\mathfrak{c}}$ and define $\psi'_i$ from the proposition. Then
\begin{equation}\label{eq2.13}
\sum_{q=1}^\infty \psi'_{1}(q)\cdots\psi'_{n}(q)
\leq \sum_{q=1}^\infty \psi_{1}(q)\cdots\psi_{n}(q)
+\sum_{q=1}^\infty q^{-1-\mathfrak{c}}
<\infty
\end{equation}
and
$$
\mathcal{S}_n(\psi_1,\dots,\psi_n)
\subset \mathcal{S}_n(\psi_1',\dots,\psi_n').
$$
Hence we may work with the set
$\mathcal{S}_n(\psi_1',\dots,\psi_n')$
instead of $\mathcal{S}_n(\psi_1,\dots,\psi_n)$,
or equivalently we may assume \eqref{eq: prod is greater than}.

Let $\psi_1,\dots,\psi_n$ be approximation functions satisfying the divergence sum condition in Theorem~\ref{thm: dream}. Then
\begin{equation}\label{eq3.20}
\sum_{t=1}^\infty 2^t\psi_1(2^t)\cdots\psi_n(2^t)=\infty.
\end{equation}
As explained in \S \ref{sec: con on psi div}, we assume $\psi_i$ satisfy \eqref{eq: psi chain}, without loss of generality.

Fix any $0<s'<\frac{1}{n-1}$. Then, by \eqref{eq3.20}, there exists a strictly increasing sequence of positive integers $(t_k)_{k\in\N}$ such that
\begin{equation}\label{eq:lower bound on prod}
\psi_{1}(q)\cdots \psi_{n}(q) q>q^{-\left(\frac{1}{n-1}-s'\right)}\qquad\text{for each } q=2^{t_k}
\end{equation}
and
\begin{equation}\label{eq3.22}
\sum_{k=1}^\infty 2^{t_k}\psi_1(2^{t_k})\cdots \psi_n(2^{t_k})=\infty.
\end{equation}
In particular, we have that for all $k\in\N$
\begin{equation}\label{eq: standing 2 in 2^t}
2^{t_k}\psi_{2}(2^{t_k})\cdots \psi_n(2^{t_k})>2^{st_k},\qquad s:=\frac{n-1}{n}s'.
\end{equation}
Define $\mathcal{T}:=\{t_k:k\in\N\}$, which is thus an infinite collection. Split $\mathcal{T}$ into two sub-collections:
$$\mathcal{T}_1:=\{t\in\mathcal{T}:
2^{t}\psi_{1}(2^{t})\cdots \psi_n(2^{t})<1\}\qquad\text{and}\qquad
\mathcal{T}_2=\mathcal{T}\setminus \mathcal{T}_1.
$$

From Lemma \ref{lem: ubiqui is enough}, it is enough to show that one of the sets on the left side of \eqref{eq: enough} has \textit{full} measure. In what follows $\brho=(\rho_i)_{i=1}^d\Rp^d\to\Rp$.
By the definition of $\Delta$ in \eqref{def: Delta},
$$\begin{aligned}\Delta(2^t, \tfrac12(\psi_{d+j}(2^t))_{j=1}^m, B, \brho(2^t)):=\bigcup_{(q,\ba,\bb)\in \mathcal{R}(2^t, (\psi_{d+j}(2^t))_{j=1}^m, B)} \tDelta(\vv a/q,\brho(2^t)),
\end{aligned}
$$
for any ball $B\subset B_0,$ where $B_0$ be as in Proposition \ref{thm: ubiq main}.
Now note that $$\begin{aligned}
 \bigcup_{(q,\ba,\bb)\in \mathcal{R}(2^t, (\psi_{d+j}(2^t))_{j=1}^m, B)} \tDelta(\vv a/q,\brho(2^t))\cap B
 \subset \bigcup_{2^{t-1}\leq q\leq 2^{t}, (q,\ba)\in J }\tDelta(\vv a/q,\brho(2^t))\cap B.\end{aligned}$$
By \eqref{eq3.22}, there is $i\in\{1,2\}$ such that
\begin{equation}\label{eq3.25}
\sum_{t\in \mathcal{T}_i} 2^t \psi_1(2^t)\cdots\psi_n(2^t)=\infty,
\end{equation}

\begin{equation}\label{mainsum}
\sum_{q=1}^\infty\psi_1(q)\cdots \psi_n(q)
\end{equation}

\medskip

\begin{theorem}[\textsc{Multiplicative convergence for manifolds}]\label{thm:mult0}Let $\mathcal{M}$ be any nondegenerate submanifold of $\mathbb{R}^n$ and $\psi:\mathbb{R}_+\to(0,1)$ be non-increasing. Then almost all points on $\mathcal{M}$ are not multiplicatively
$\psi$-approximable if
$$
\sum_{q=1}^\infty \psi(q) (\log q)^{n-1}<\infty.
$$
\end{theorem}

Theorem \ref{thm: dream0} is first proved for the case when the manifold is parametrized in the so-called Monge form in \S \ref{sec: div} (for divergence) and \S \ref{sec: convergence} (for convergence). The proof is then generalized to arbitrary  nondegenerate manifolds in \S \ref{sec: general}. The proof of  Theorem~\ref{thm:mult0} is presented in \S \ref{sec: multi}.

\subsection{Previous results and comparison}

We first discuss the unweighted case ($\psi_1=\cdots=\psi_n$) of Theorem~\ref{thm: dream0}. In this setting, Theorem~\ref{thm: dream0} was first established for nondegenerate $C^3$ planar curves ($n=2$) in \cite{BDV07} in the divergence case and in \cite{VV06} in the convergence case. The $C^3$ assumption was later removed in \cite{BZ10} (divergence) and \cite{Hua15} (convergence). In higher dimensions, the divergence case was proved by the first-named author in \cite{Ber12} for analytic nondegenerate manifolds and was recently extended to all nondegenerate manifolds in \cite{BD}, having previously been known only for nondegenerate curves \cite{BVVZ21}. The convergence case was settled in \cite{BY} for all nondegenerate manifolds.

Much less is known in the weighted case, which allows the approximation functions $\psi_1,\dots,\psi_n$ to be different. The story again begins with planar curves, studied in \cite{BV07}, where the authors proved both the convergence and divergence cases of Theorem~\ref{thm: dream0}: the divergence case for $C^3$ curves and the convergence case for rational quadrics. The latter was later extended to arbitrary $C^3$ nondegenerate planar curves in \cite{Bad_Lev_07}.

In higher dimensions, the general problem was posed in \cite[Problem~S2]{BV07} as follows.

\smallskip

\noindent\textbf{Problem~1:} Given a nondegenerate manifold $\mathcal{M}$ and approximation functions $\psi_1,\dots,\psi_n$, find the weakest condition under which $\mathcal{S}_n(\psi_1,\dots,\psi_n)\cap\mathcal{M}$ has Lebesgue measure zero.

\smallskip

Working towards this general problem, Srivastava \cite{Srivastava2025} recently established the convergence case of Theorem~\ref{thm: dream0} for a subclass of nondegenerate manifolds with a curvature condition, under the additional constraint $\psi_1=\cdots=\psi_d$ on the approximation functions. To the best of our knowledge, the divergence part of Theorem~\ref{thm: dream0} for $n>2$ and arbitrary approximation functions $\psi_1,\dots,\psi_n$ has not been previously addressed. Theorem~\ref{thm: dream0} resolves Problem~1 in full and, at the same time, generalises the results of \cite{BY} and \cite{BD} from equal to arbitrary approximation functions $\psi_1,\dots,\psi_n$.

\medskip

We now turn to the multiplicative case, concerning the set $\cSM_n(\psi)$. In this setting, progress to date has been far more limited, and even for planar curves the theory remains incomplete. Indeed, for nondegenerate planar curves, Theorem~\ref{thm:mult0} was proved in \cite{Hua15}, having been previously established for rational quadrics \cite{BV07} and under the $C^3$ assumption \cite{Bad_Lev_07}.

As with Problem~1, the general question regarding $\cSM_n(\psi)$ was posed in \cite[Problem~S2]{BV07}:

\smallskip

\noindent\textbf{Problem~2:} Given a nondegenerate manifold $\mathcal{M}$ and an approximation function $\psi$, find the weakest condition under which $\cSM_n(\psi)\cap\mathcal{M}$ has Lebesgue measure zero.

\smallskip

Theorem~\ref{thm:mult0} contributes towards the resolution of this problem by providing an analogue of the convergence case of Theorem~G for all nondegenerate manifolds.

A natural next step would be to prove the divergence case of Theorem~G for nondegenerate manifolds. This remains a major open problem and has so far been addressed only for lines in $\mathbb{R}^2$ \cite{MR4068301, MR3813593, MR4701882} and for ``vertical lines'' in higher dimensions \cite{MR4016056, MR4706444}. Even these limited cases required substantial new ideas, including the development of a Bohr sets technique \cite{MR3813593} and an effective asymptotic equidistribution method \cite{MR4701882}.

Finally, we note that our results may be viewed as complete analogues of the corresponding results for dual approximation on manifolds. In that setting, the convergence case for both weighted and multiplicative approximation was established by Bernik, Kleinbock and Margulis \cite{BKM}, while the divergence case for weighted approximation was proved in \cite{MR2989975}.

\subsection{Further remarks}

For a manifold $\mathcal{M} \subset \mathbb{R}^n$ of dimension $d$ to be nondegenerate, it is necessary that $\mathcal{M}$ be $C^l$ with ${d+l \choose l} \ge n+1$. For example, if $d=2$, this reduces to $l^2 + 3l \ge 2n$. Clearly there exist manifolds that are just $C^l$ for the smallest $l$ satisfying this inequality. The above theorems use this minimal smoothness, sufficient to meet the nondegeneracy condition.

The proof of Theorem~\ref{thm: dream0} builds on the linearization techniques developed in \cite{BY} and \cite{BD}. The new features include linearization in different coordinates with different rates, including linearization in a single direction only. For the divergence case, we use the recent \textit{ubiquity} result of Kleinbock and Wang \cite{KWang23}. To construct an appropriate ubiquitous system, we provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \cite{BKM} that enables independent control of different partial derivatives (see \S \ref{new BKM}). At the technical level, our proof differs from the previous argument even in the \textit{unweighted} case in \cite{BD}, offering a new perspective.

Finally, we note that, while preparing this paper, we became aware of related work by Chow, Srivastava, Technau, and Yu \cite{MultCSTY} concerning Problem~2. Using a different approach initiated in \cite{SST}, they establish a multiplicative convergence result for \(C^\infty\) nondegenerate manifolds and a natural class of affine subspaces.
We emphasize that both Problem~1 and Problem~2 remain of interest and are still widely open for degenerate manifolds, such as affine subspaces and submanifolds that are nondegenerate relative to an affine subspace, as considered in \cite{MR1982150}.

Theorem~\ref{thm: dream0} is proved first “for the case when the manifold is parametrized in the so-called Monge form” (\S\,\ref{sec: div} for divergence and \S\,\ref{sec: convergence} for convergence), and “then generalized to arbitrary nondegenerate manifolds” in \S\,\ref{sec: general}. The proof “builds on the linearization techniques developed in \cite{BY} and \cite{BD},” with “new features” including “linearization in different coordinates with different rates, including linearization in a single direction only.” For the divergence case, the argument uses “the recent \textit{ubiquity} result of Kleinbock and Wang \cite{KWang23}.” To “construct an appropriate ubiquitous system,” the authors “provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \cite{BKM} that enables independent control of different partial derivatives” (see \S\,\ref{new BKM}). They also note that “at the technical level, our proof differs from the previous argument even in the \textit{unweighted} case in \cite{BD}, offering a new perspective.”

To prove the main theorem, the authors proceed first in the case when the manifold is parametrized in the so-called Monge form (for divergence they argue earlier, and for convergence they argue later), and then generalize to arbitrary nondegenerate manifolds later. The proof builds on the linearization techniques developed in

\subsection{Quantitative Nondivergence}\label{new BKM}

Our proofs will require an effective measure estimate for the following set
\begin{align}
\mathfrak{S}_{F}&(\delta,\bK,\bT)=\nonumber\\[1ex]
&=\left\{\bx\in \bU:\exists\;(a_0,\ba)\in\Z\times\Z^n_{\neq\bf0}\;\;\text{such that }\left.
\begin{array}{l}
|a_0+F(\bx)\ba^\top|<\delta\\[1ex]
\vert \partial_iF(\bx)\ba^\top\vert<K_i\;\;(1\le i\le d) \\[1ex]
\vert a_k\vert<T_k ~~(1\leq k\leq n)
\end{array}
\right.\right\},\label{eq: new set}
\end{align}
where $\ba=(a_1,\dots,a_n)$, $\bT=(T_1,\dots,T_n)$, $\bK=(K_1,\dots,K_d)$, and $\delta,K_1,\cdots,K_d,T_1,\cdots, T_n$ are positive real parameters.

The following result represents a generalisation of \cite[Theorem 1.4]{BKM}
and appears in \cite[Theorem 3.1]{BD} when $K_1=\dots=K_d$. Its proof in the general case follows the same lines as that of \cite[Theorem 3.1]{BD} with minor modifications which will be explained below.

\begin{theorem}\label{BKM_new2}
Let $F:\bU\to\R^n$ be as defined in \eqref{eq: F}, $l$-nondegenerate at $\bx_0$.
Then there exists a ball $B_0\subset\bU$ centred at $\bx_0$ and a constant $E>0$ depending on $B_0$ and $F$ such that for any choice of $\delta,K_1,\cdots,K_d,T_1,\cdots, T_n$ satisfying
\begin{equation}\label{eqn5.2}
0<\delta\le 1,\qquad T_1,\dots,T_n\ge1,\qquad K_1,\cdots,K_d>0 \quad\text{and}\quad \delta^{n}<K \frac{T_1\cdots T_n}{\max_i T_i},
\end{equation}

as well as on new features including linearization in different coordinates with different rates, including linearization in a single direction only. For the divergence case, the argument uses the recent ubiquity result of Kleinbock and Wang \cite{KWang23}. To construct an appropriate ubiquitous system, the authors provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \cite{BKM} that enables independent control of different partial derivatives (as in the quantitative nondivergence subsection above). They also note that at the technical level, their proof differs from the previous argument even in the unweighted case in \cite{BD}, offering a new perspective.

[\textsc{Weighted Khintchine for manifolds}]\label{thm: dream0}
Let $\mathcal{M}$ be any nondegenerate submanifold of $\mathbb{R}^n$ and $\psi_1,\dots,\psi_n:\mathbb{R}_+\to(0,1)$ be non-increasing. Then almost all points on $\mathcal{M}$ are $(\psi_1,\dots,\psi_n)$-approximable if the series
\begin{equation}\label{mainsum}
\sum_{q=1}^\infty\psi_1(q)\cdots \psi_n(q)
\end{equation}
diverges, and are not $(\psi_1,\dots,\psi_n)$-approximable if the equation above converges.

Biconditional or Equivalence
Implication

Question:
Let \(\mathcal M\subset \mathbb R^n\) be a nondegenerate submanifold of \(\mathbb R^n\) (in the standard metric Diophantine approximation sense), and let \(\psi_1,\dots,\psi_n:\mathbb R_+\to(0,1)\) be non-increasing functions. A point \(\mathbf x=(x_1,\dots,x_n)\in\mathbb R^n\) is called \((\psi_1,\dots,\psi_n)\)-approximable if there exist infinitely many tuples \((q,p_1,\dots,p_n)\in\mathbb N\times\mathbb Z^n\) such that
\[
|q x_i-p_i|<\psi_i(q)\qquad (1\le i\le n).
\]
Which statement about the set of points of \(\mathcal M\) with this property is valid, where “almost all” is with respect to the natural Lebesgue measure on \(\mathcal M\)?

If
\[
\sum_{q=1}^\infty \psi_1(q)\cdots \psi_n(q)=\infty,
\]
then almost all points on \(\mathcal M\) are \((\psi_1,\dots,\psi_n)\)-approximable; and if
\[
\sum_{q=1}^\infty \psi_1(q)\cdots \psi_n(q)<\infty,
\]
then almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable.

Choices:
B.
If
\[
\sum_{q=1}^\infty \psi_1(q)\cdots \psi_n(q)=\infty,
\]
then almost all points on \(\mathcal M\) are \((\psi_1,\dots,\psi_n)\)-approximable; and if
\[
\sum_{q=1}^\infty q\,\psi_1(q)\cdots \psi_n(q)<\infty,
\]
then almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable.
C.
If
\[
\sum_{q=1}^\infty \psi_1(q)\cdots \psi_n(q)<\infty,
\]
then almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable.
D.
If
\[
\sum_{q=1}^\infty \psi_1(q)\cdots \psi_n(q)=\infty,
\]
then every point of \(\mathcal M\) is \((\psi_1,\dots,\psi_n)\)-approximable; and if
\[
\sum_{q=1}^\infty \psi_1(q)\cdots \psi_n(q)<\infty,
\]
then almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable.
E.
If for each \(i\) one has
\[
\sum_{q=1}^\infty \psi_i(q)=\infty,
\]
then almost all points on \(\mathcal M\) are \((\psi_1,\dots,\psi_n)\)-approximable; and if for some \(i\) one has
\[
\sum_{q=1}^\infty \psi_i(q)<\infty,
\]
then almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable.

critical summand uses product \(\psi_1\cdots\psi_n\) with no extra \(q\)-factor

dropped the divergence/full-measure direction

almost-all conclusion strengthened to every point

product criterion replaced by coordinatewise series criteria


Question:
Let \(\mathcal M\subset\mathbb R^n\) be a nondegenerate submanifold, and let \(\psi_1,\dots,\psi_n:\mathbb R_+\to(0,1)\) be non-increasing functions. A point \(\mathbf x=(x_1,\dots,x_n)\in\mathbb R^n\) is called \((\psi_1,\dots,\psi_n)\)-approximable if there exist infinitely many \((q,p_1,\dots,p_n)\in\mathbb N\times\mathbb Z^n\) such that
\[
|q x_i-p_i|<\psi_i(q)\qquad(1\le i\le n).
\]
Which statement gives the correct measure-theoretic criterion for the set of \((\psi_1,\dots,\psi_n)\)-approximable points on \(\mathcal M\)?

Almost all points on \(\mathcal M\) are \((\psi_1,\dots,\psi_n)\)-approximable if \(\sum_{q=1}^\infty \psi_1(q)\cdots\psi_n(q)=\infty\), and almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable if \(\sum_{q=1}^\infty \psi_1(q)\cdots\psi_n(q)<\infty\).

Choices:
B. Almost all points on \(\mathcal M\) are \((\psi_1,\dots,\psi_n)\)-approximable if \(\sum_{q=1}^\infty q\,\psi_1(q)\cdots\psi_n(q)=\infty\), and almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable if \(\sum_{q=1}^\infty q\,\psi_1(q)\cdots\psi_n(q)<\infty\).
C. If \(\sum_{q=1}^\infty \psi_1(q)\cdots\psi_n(q)<\infty\), then almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable.
D. There exists a constant \(E>0\), depending only on \(\mathcal M\), such that almost all points on \(\mathcal M\) are \((\psi_1,\dots,\psi_n)\)-approximable whenever \(\sum_{q=1}^\infty \psi_1(q)\cdots\psi_n(q)=\infty\) and \(\psi_i(q)\le E\,q^{-1}\) for all \(q\) and all \(1\le i\le n\); and almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable if \(\sum_{q=1}^\infty \psi_1(q)\cdots\psi_n(q)<\infty\).
E. Almost all points on \(\mathcal M\) are \((\psi_1,\dots,\psi_n)\)-approximable if for every sufficiently large \(q\) one has \(q\,\psi_1(q)\cdots\psi_n(q)\ge 1\), and almost all points on \(\mathcal M\) are not \((\psi_1,\dots,\psi_n)\)-approximable if \(\sum_{q=1}^\infty \psi_1(q)\cdots\psi_n(q)<\infty\).