Badly approximable points for diagonal approximation in solenoids
Volume 199 / 2021
Acta Arithmetica 199 (2021), 153-161
MSC: 11J61, 11J83.
DOI: 10.4064/aa200425-7-12
Published online: 2 March 2021
Abstract
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.
