Critical bases for ternary alphabets

TL;DR

This paper explores the structure of unique expansions in non-integer bases for ternary alphabets, extending known results from binary cases and identifying critical base values where the nature of unique expansions changes.

Contribution

It generalizes the concept of critical bases for unique expansions from binary to ternary alphabets, providing new insights into their properties.

Findings

Identifies a critical base for ternary alphabets analogous to the binary case.

Shows the transition from countably many to continuum of unique expansions at the critical base.

Extends the understanding of non-integer base expansions to larger alphabets.

Abstract

Glendinning and Sidorov discovered an important feature of the Komornik-Loreti constant $q'\approx1.78723$ in non-integer base expansions on two-letter alphabets: in bases $1<q<q'$ only countably numbers have unique expansions, while for $q\ge q'$ there is a continuum of such numbers. We investigate the analogous question for ternary alphabets.