Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3

TL;DR

This paper investigates the generalized Hausdorff dimension of specific irrationals with a Lagrange value of exactly 3, revealing precise dimension thresholds and classifying their topological structure.

Contribution

It precisely determines the critical dimension function where the generalized Hausdorff measure of these irrationals transitions from infinite to zero, and classifies their topological properties.

Findings

Exact cutoff point for generalized Hausdorff dimension identified.

The measure is either zero or not σ-finite, depending on the dimension function.

Attainable and non-attainable subsets have different generalized Hausdorff dimensions.

Abstract

We study the generalized Hausdorff dimension of some natural subsets of $k^{-1}(3)$, where $k^{-1}(3)$ consists of the real numbers $x$ for which $\left| x-\frac{p}{q} \right|<\frac{1}{(3+\varepsilon)q^2}$ has infinitely many rational solutions $\frac{p}{q}$ for any $\varepsilon<0$ but only finitely many for any $\varepsilon>0$. It is well known that $k^{-1}(3)$ is an uncountable set with Hausdorff dimension zero. Given any dimension function $h$, we determine the exact "cut point" at which the generalized Hausdorff dimension $\mathcal{H}^h(k^{-1}(3))$ drops from infinity to zero. In particular we show that such a measure is always zero or not $\sigma$--finite, and, as an application, we can classify topologically $k^{-1}(3)$. Moreover, we show that the subset of attainable elements of $k^{-1}(3)$ has the same generalized Hausdorff dimension as $k^{-1}(3)$, but the subset of non--attainable elements of $k^{-1}(3)$ has a "strictly smaller" generalized Hausdorff dimension.