2013/05/01 by Jayadev S. Athreya, Athreya, Jayadev S., Anish Ghosh +3
Mathematics · #11J70 #11K60 #37A17 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #math.DS #math.NT #msc:11J70 #msc:11K60 #msc:37A17
paper · pdf · doi:10.48550/arxiv.1305.0296
20 pages, 1 figure. Noteworthy changes from the previous version: New title. New result added (Theorem 1.1). Strengthening of Theorem 1.3
arxiv created 2014/11/26 · arxiv updated 2014/11/27
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 Strömbergsson \citeMS3 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 \citeEMM and Kleinbock-Margulis \citeKM and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.