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(\tau)={x\in\mathbb{R}: |qx| < |q|^{-\tau} \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 $\mathbb{R}^n, n\geq 2,$ the set of badly approximable vectors is not Salem.