Hausdorff dimension and exact approximation order in $\mathbb{R}^n$

TL;DR

This paper investigates the Hausdorff dimension of sets of points in b^n that are exactly b-approximable, providing new answers to longstanding questions in Diophantine approximation.

Contribution

It establishes the non-emptiness and computes the Hausdorff dimension of exactly b-approximable sets in b^n, resolving open problems for dimensions nbb2.

Findings

Non-empty sets of exactly b-approximable points in b^n.

Explicit Hausdorff dimension formulas for these sets.

Resolution of questions posed by Jarnedk and others.

Abstract

Given a non-increasing function $\psi\colon\mathbb{N}\to\mathbb{R}^+$ such that $s^{\frac{n+1}{n}}\psi(s)$ tends to zero as $s$ goes to infinity, we show that the set of points in $\mathbb{R}^n$ that are exactly $\psi$-approximable is non-empty, and we compute its Hausdorff dimension. For $n\geq 2$, this answers questions of Jarn\'{i}k and of Beresnevich, Dickinson, and Velani.