Spiraling of approximations and spherical averages of Siegel transforms

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 $\text{every}$ 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 Strömbergsson \cite{MS3} to show a spherical average result for Siegel transforms on ${\mathrm{SL}}_{d+1}(\mathbb{R})/{\mathrm{SL}}_{d+1}(\mathbb{Z})$. 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.

arXiv comment20 pages, 1 figure. Noteworthy changes from the previous version: New title. New result added (Theorem 1.1). Strengthening of Theorem 1.3

References in corpus (1)

• Spherical averages of Siegel transforms for higher rank diagonal actions and applications