Expanding maps and continued fractions
Number Theory
2015-02-10 v2 Dynamical Systems
Abstract
We obtain a power saving in the error term for a semigroup congruence lattice point count related to continued fractions. This is done by adapting arguments from recent work of Oh and Winter (2014) that give uniform bounds for certain transfer operators in the congruence aspect. Our arguments also build crucially on work of Naud (2005) and Bourgain, Gamburd and Sarnak (2011). The result we obtain, together with a certain conjecture about the multiplicative combinatorics of that we highlight in the sequel, can be used to obtain an improvement on the size of the exceptional set in Bourgain and Kontorovich's work (2014) on Zaremba's conjecture.
Cite
@article{arxiv.1412.4284,
title = {Expanding maps and continued fractions},
author = {Michael Magee and Hee Oh and Dale Winter},
journal= {arXiv preprint arXiv:1412.4284},
year = {2015}
}