Spiraling of approximations and spherical averages of Siegel transforms
Journal of the London Mathematical Society
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  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  and KleinbockMargulis  and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed
J.S.A. partially supported by NSF grant DMS 1069153, and NSF grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). A.G. partially supported by the Royal Society. J.T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147.
This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.
Vol. 91, pp. 383 - 404