Effective Ratner theorem for ASL(2,R) and gaps in \sqrt{n} modulo 1
Dynamical Systems
2016-02-29 v4 Number Theory
Abstract
Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Str\"ombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of \sqrt{n} modulo 1.
Keywords
Cite
@article{arxiv.1311.6387,
title = {Effective Ratner theorem for ASL(2,R) and gaps in \sqrt{n} modulo 1},
author = {Tim Browning and Ilya Vinogradov},
journal= {arXiv preprint arXiv:1311.6387},
year = {2016}
}
Comments
24 pages; further extensive rewriting of section 5