On the continuity of the Hausdorff dimension of the univoque set
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2018
- DOI:
- 10.48550/arXiv.1804.02879
- arXiv:
- arXiv:1804.02879
- Bibcode:
- 2018arXiv180402879A
- Keywords:
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- Mathematics - Dynamical Systems;
- Primary: 11A63;
- Secondary: 37B10;
- 28A78
- E-Print:
- 18 pages