Expansions in multiple bases

TL;DR

This paper introduces a systematic study of real number expansions in multiple bases, generalizing classical single-base expansions and exploring various types like greedy and unique expansions in this new context.

Contribution

It is the first to analyze multiple-base expansions systematically, extending classical expansion concepts to a broader, more general setting.

Findings

Defined and analyzed greedy, quasi-greedy, lazy, quasi-lazy, and unique expansions in multiple bases.

Established foundational properties and relationships among different types of expansions in this setting.

Provided initial results and frameworks for future research in multi-base number expansions.

Abstract

Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy, lazy, quasi-lazy and unique expansions in multiple bases.