An exponentially shrinking problem

2023/02/21 by Hussain, Mumtaz, Shi, Junjie
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2302.10379

Abstract

The Jarník-Besicovitch theorem is a fundamental result in metric number theory which gives the Hausdorff dimension for limsup sets. We investigate a related problem of estimating the Hausdorff dimension of a liminf set. Let h>0, τ≥ 1, and for any j≥ 1 define the integer sequence q_j+1=q_j^h. We prove the Hausdorff dimension of the set Λ^\bftheta_d(τ)=\\xx∈[0, 1)^d: ‖q_jx_i-θ_i‖

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