Expansions in multiple bases over general alphabets

TL;DR

This paper develops a comprehensive framework for expansions in multiple bases over general alphabets, extending existing theories and improving results on unique expansions.

Contribution

It introduces a unified alphabet-base system framework that encompasses previous models and extends Parry's lexicographic theory to this broader setting.

Findings

Unified framework for alphabet-base expansions

Extension of Parry's lexicographic theory

Generalization of results on unique expansions

Abstract

Expansions in non-integer bases have been extensively investigated since a pioneering work of R\'enyi. We introduce a more general framework of alphabet-base systems that also includes Pedicini's general alphabets and the multiple-base expansions of Neunh\"auserer and Li. We extend the Parry type lexicographic theory to this setup, and we improve and generalize various former results on unique expansions.