Hausdorff Dimension of Growth Rate Level Sets in $\theta$-expansions

TL;DR

This paper studies the Hausdorff dimension of level sets defined by digit growth rates in $ heta$-expansions, extending previous results from regular continued fractions to a broader class.

Contribution

It proves that these level sets have full Hausdorff dimension for any specified growth rate, generalizing earlier work to $ heta$-expansions.

Findings

Level sets have full Hausdorff dimension for all growth rates.

Extension of previous results from regular continued fractions to $ heta$-expansions.

Constructs explicit subsets with controlled digit growth to establish dimension.

Abstract

We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \geq 0$, we prove that the set \[ E_\theta(\alpha) = \left\{ x \in [0, \theta] \setminus \mathbb{Q} : \lim_{n \to {+}\infty} \frac{L_{n,\theta}(x) \log n \log \log n}{S_{n,\theta}(x) - L_{n,\theta}(x)} = \alpha \right\} \] has full Hausdorff dimension. This extends previous work of Zhang and {L\"u} (2016) on regular continued fractions to the broader framework of $\theta$-expansions. The proof involves constructing explicit subsets with controlled digit growth and establishing dimension preservation through H\"older-continuous mappings.