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arXiv · 1804.02879

On the continuity of the Hausdorff dimension of the univoque set

Pieter Allaart·Derong Kong
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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.

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Pieter Allaart, Derong Kong. 2018-04-09. On the continuity of the Hausdorff dimension of the univoque set. https://doi.org/10.1016/j.aim.2019.106729

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