Non-Salem Sets in Metric Diophantine Approximation

Abstract

A classical result of Kaufman states that, for each τ > 1, the set of τ-well approximable numbers {equation presented} is a Salem set. A natural question to ask is whether the same is true for the sets of τ -well approximable n × d matrices when nd > 1 and t > d/n. We prove the answer is no by computing the Fourier dimension of these sets. In addition, we show that the set of badly approximable n × d matrices is not Salem when nd > 1. The case of nd = 1, that is, the badly approximable numbers, remains unresolved.