Expanding maps and continued fractions

Michael Magee, Hee Oh, Dale Winter

Published 2014-12-13Version 1

We obtain a power saving in the error term for a semigroup congruence lattice point count related to continued fractions, which was an important missing ingredient for a power savings error term in Bourgain-Kontorovich's work (2014) on Zaremba's conjecture. 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).