Pointwise equidistribution with an error rate and with respect to unbounded functions

TL;DR

This paper advances the understanding of pointwise equidistribution in homogeneous spaces by providing explicit error rates for smooth functions and establishing equidistribution for certain unbounded functions, with applications in number theory.

Contribution

It introduces an effective error rate for equidistribution of trajectories and extends equidistribution results to unbounded functions like Siegel transforms, enhancing previous work.

Findings

Error rate for equidistribution with smooth functions

Equidistribution for unbounded Siegel transforms

Number-theoretic applications to lattice approximation spiraling

Abstract

Consider $G=\SL_{ d }(\mathbb R)$ and $ \Gamma=\SL_{ d }(\mathbb Z)$. It was recently shown by the second-named author \cite{s} that for some diagonal subgroups $\{g_t\}\subset G$ and unipotent subgroups $U\subset G$, $g_t$-trajectories of almost all points on all $U$-orbits on $G/\Gamma$ are equidistributed with respect to continuous compactly supported functions $\varphi$ on $G/\Gamma$. In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when $\varphi$ is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on $\R^d$. For the first part we use a method based on effective double equidistribution of $g_t$-translates of $U$-orbits, which generalizes the main result of \cite{km12}. The second part is based on Schmidt's results on counting of lattice points. Number-theoretic consequences involving spiraling of lattice approximations, extending recent work of Athreya, Ghosh and Tseng \cite{agt1}, are derived using the equidistribution result.