Bohr sets and multiplicative diophantine approximation
Number Theory
2018-07-18 v1 Combinatorics
Abstract
In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.
Cite
@article{arxiv.1703.07016,
title = {Bohr sets and multiplicative diophantine approximation},
author = {Sam Chow},
journal= {arXiv preprint arXiv:1703.07016},
year = {2018}
}