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arXiv · 1305.0296

Spiraling of approximations and spherical averages of Siegel transforms

Jayadev S. Athreya·Anish Ghosh·Jimmy Tseng
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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.

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Jayadev S. Athreya, Anish Ghosh, Jimmy Tseng. 2013-05-01. Spiraling of approximations and spherical averages of Siegel transforms. https://arxiv.org/abs/1305.0296

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