Non-Salem sets in metric Diophantine approximation

Abstract

A classical result of Kaufman states that, for each $\tau >1,$ the set of well approximable numbers \[ E(τ)=\{x\in\mathbb{R}: \|qx\| < |q|^{-τ} \text{ for infinitely many integers q}\} \] is a Salem set with Hausdorff dimension $2/(1+\tau )$. A natural question to ask is whether the same phenomena holds for well approximable vectors in ${\mathbb{R}}^n.$ We prove that this is in general not the case. In addition, we also show that in ${\mathbb{R}}^n,n\ge 2,$ the set of badly approximable vectors is not Salem.