Spiraling of approximations and spherical averages of Siegel transforms

TL;DR

This paper investigates how approximations to vectors in Euclidean space spiral around and distribute on the sphere, providing almost everywhere results and demonstrating uniform distribution for all lattices, with elementary techniques and explicit counterexamples.

Contribution

It establishes new spherical average results for Siegel transforms and shows uniform distribution of approximation directions for all unimodular lattices, extending previous work with elementary methods.

Findings

Almost everywhere spiraling behavior of approximations

Uniform distribution of approximation directions for all lattices

Existence of counterexamples with non-uniform distribution

Abstract

We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for $\textit{every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Str\"ombergsson \cite{MS3} to show a spherical average result for Siegel transforms on $\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.