Non-Salem sets in metric Diophantine approximation

TL;DR

This paper investigates the properties of well approximable and badly approximable sets in higher dimensions, showing they are generally not Salem sets, contrasting with classical one-dimensional results.

Contribution

It demonstrates that in dimensions two and higher, well approximable and badly approximable sets do not exhibit Salem set properties, unlike the classical one-dimensional case.

Findings

Well approximable sets in higher dimensions are not Salem sets.

Badly approximable vectors in higher dimensions are not Salem.

Contrasts classical one-dimensional Salem set results.

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.