Non-Salem Sets in Metric Diophantine Approximation

Abstract

Abstract A classical result of Kaufman states that, for each $\tau>1$, the set of $\tau $-well approximable numbers $$ \begin{align*} & E(\tau)=\{x \in \mathbb{R}: |xq-r| < |q|^{-\tau} \text{ for infinitely many integer pairs } (q,r)\} \end{align*}$$is a Salem set. A natural question to ask is whether the same is true for the sets of $\tau $-well approximable $n \times d$ matrices when $nd>1$ and $\tau> d/n$. We prove the answer is no by computing the Fourier dimension of these sets. In addition, we show that the set of badly approximable $n \times d$ matrices is not Salem when $nd> 1$. The case of $nd=1$, that is, the badly approximable numbers, remains unresolved.