On the continuity of the Hausdorff dimension of the univoque set

TL;DR

This paper provides a new combinatorial proof demonstrating the continuity of the Hausdorff dimension of the univoque set in non-integer bases, resolving a gap in previous research and confirming a long-standing conjecture.

Contribution

It introduces a novel combinatorial method to prove the continuity of Hausdorff dimension for the univoque set, replacing the previous complex proof.

Findings

Confirmed the continuity of Hausdorff dimension in base q

Provided a new combinatorial proof approach

Resolved a gap in prior mathematical literature

Abstract

In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that this Hausdorff dimension is continuous in $q$. Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach.