Spiraling of approximations and spherical averages of Siegel transforms
Athreya, JS; Ghosh, A; Tseng, J
Date: 1 April 2015
Journal
Journal of the London Mathematical Society
Publisher
Wiley
Publisher DOI
Abstract
We consider the question of how approximations satisfying Dirichlet’s theorem spiral
around vectors in Rd. We give pointwise almost everywhere results (using only the Birkhoff ergodic
theorem on the space of lattices). In addition, we show that for every unimodular lattice, on average,
the directions of approximates spiral in a ...
We consider the question of how approximations satisfying Dirichlet’s theorem spiral
around vectors in Rd. We give pointwise almost everywhere results (using only the Birkhoff ergodic
theorem on the space of lattices). In addition, we show that for 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 Strombergsson [19]
to show a spherical average result for Siegel transforms on SLd+1(R)/ SLd+1(Z). Our techniques
are elementary. Results like this date back to the work of Eskin-Margulis-Mozes [9] and KleinbockMargulis
[14] and have wide-ranging applications. We also explicitly construct examples in which
the directions are not uniformly distributed
Mathematics and Statistics
Faculty of Environment, Science and Economy
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