Badly approximable points for diagonal approximation in solenoids
arXiv:2004.12153
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.
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)