Hausdorff distance of univoque sets

TL;DR

This paper studies the continuity of the Hausdorff distance between univoque sets in non-integer base expansions, revealing complex structural properties of these unique expansion sets.

Contribution

It investigates the Hausdorff continuity properties of univoque sets, providing new insights into their geometric and topological structure.

Findings

Univoque sets exhibit intricate Hausdorff continuity behavior.

The structure of unique expansion sets is more complex than in integer bases.

Results contribute to understanding the geometric properties of non-integer base expansions.

Abstract

Expansions in non-integer bases have been investigated abundantly since their introduction by R\'enyi. It was discovered by Erd\H{o}s et al. that the sets of numbers with a unique expansion have a much more complex structure than in the integer base case. The present paper is devoted to the continuity properties of these maps with respect to the Hausdorff metric.