Non-Salem sets in multiplicative Diophantine approximation

2024/09/19 by Tan, Bo, Zhou, Qing-Long
#11J83 #11K55 #28A80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2409.12557

Abstract

In this paper, we answer a question of Cai-Hambrook in (arXiv\colon 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative ψ-well approximable set M₂^×(ψ)=\(x₁,x₂)∈ [0,1]²\colon ‖qx₁‖‖qx₂‖lt;ψ(q) for infinitely many q∈ \N\, where ψ\colon\N→ [0,(1)/(4)) is a positive function satisfying ∑_qψ(q)log(1)/(ψ(q))<∞. As a corollary, we show that the set M₂^×(q↦ q^-τ) is non-Salem for τ>1.

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