On the continuity of the Hausdorff dimension of the univoque set

Pieter Allaart; Derong Kong

doi:10.1016/j.aim.2019.106729

paper

On the continuity of the Hausdorff dimension of the univoque set

arXiv:1804.02879 · doi:10.1016/j.aim.2019.106729

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 $U_{q}$ 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.

18 pages