Hausdorff dimension in inhomogeneous Diophantine approximation

TL;DR

This paper investigates the Hausdorff dimension of sets of numbers with specific inhomogeneous Diophantine approximation properties, linking full dimension to the average growth of partial quotients of the irrational number.

Contribution

It establishes a characterization of when the set of epsilon-badly approximable numbers has full Hausdorff dimension based on the average growth of partial quotients, including one-sided and higher-dimensional cases.

Findings

Full Hausdorff dimension occurs if and only if the average of log partial quotients tends to infinity.

Stronger results are obtained for the case when partial quotients tend to infinity.

Partial results are established for higher-dimensional inhomogeneous approximation.

Abstract

Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $\epsilon$ if and only if $\alpha$ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $\alpha$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.