Non-Salem sets in metric Diophantine approximation

2021/09/23 by Hambrook, Kyle, Yu, Han · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.11332

Abstract

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

Cited by

• The dimension of well approximable numbers

Related

• Explicit Salem sets in ℝⁿ
• Explicit Salem sets and applications to metrical Diophantine approximation
• Non-Salem sets in multiplicative Diophantine approximation
• Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the p-adic numbers
• Diophantine approximation and badly approximable sets