Badly approximable points for diagonal approximation in solenoids
Number Theory
2020-11-12 v2
Abstract
In this paper we investigate the problem of how well points in finite dimensional p-adic solenoids can be approximated by rationals. The setting we work in was previously studied by Palmer, who proved analogues of Dirichlet's theorem and the Duffin-Schaeffer theorem. We prove a complementary result, showing that the set of badly approximable points has maximum Hausdorff dimension. Our proof is a simple application of the elegant machinery of Schmidt's game.
Cite
@article{arxiv.2004.12153,
title = {Badly approximable points for diagonal approximation in solenoids},
author = {Huayang Chen and Alan Haynes},
journal= {arXiv preprint arXiv:2004.12153},
year = {2020}
}
Comments
New version: Clarified some points in proofs. Added more explanation about how to derive the dimension result from the winning property in the setting of p-adic solenoids (Section 2)