An exponentially shrinking problem

TL;DR

This paper studies the Hausdorff dimension of certain liminf sets in metric number theory, extending classical results by analyzing sets defined via exponential shrinking sequences and providing heuristics for multiplicative analogs.

Contribution

It establishes the Hausdorff dimension of a specific liminf set related to exponential sequences and offers heuristics for the multiplicative case, advancing understanding of shrinking target problems.

Findings

Hausdorff dimension of the liminf set is determined for exponential sequences.

Provides heuristics for the Hausdorff dimension of multiplicative sets.

Extends classical metric number theory results to new shrinking target scenarios.

Abstract

The Jarn\'ik-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 $h>0, \tau\geq 1$, and for any $j\geq 1$ define the integer sequence $q_{j+1}=q_j^h$. We prove the Hausdorff dimension of the set $$\Lambda^\bftheta_d(\tau)=\left\{\xx\in[0, 1)^d: \|q_jx_i-\theta_i\|<q_j^{-\tau} \ \text{for all } j\geq 1, i=1,2,\cdots,d\right\},$$ where $\left\|\star\right\|$ denotes the distance to the nearest integer and $\bftheta\in [0, 1)^d$ is fixed. We also give some heuristics for the Hausdorff dimension of the corresponding multiplicative set $$\MM_d^\bftheta(\tau)=\left\{\xx\in[0, 1)^d:\prod_{i=1}^d \|q_jx_i-\theta_i\|<q_j^{-\tau} \ \text{for all } j\geq 1\right\}.$$