Expansions in Cantor real bases

TL;DR

This paper introduces and analyzes a generalized form of number representations called $oldsymbol{eta}$-representations, extending classical base and Cantor series representations, and explores their properties, including a Parry-type characterization and conditions for sofic shifts in special cases.

Contribution

It generalizes existing number representation theories to arbitrary Cantor real bases and establishes fundamental properties, including a Parry-like theorem and criteria for soficity in periodic bases.

Findings

Generalized number representations in Cantor real bases.

Proved a Parry-type theorem for $oldsymbol{eta}$-representations.

Characterized when the $oldsymbol{eta}$-shift is sofic for periodic bases.

Abstract

We introduce and study series expansions of real numbers with an arbitrary Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$, which we call $\boldsymbol{\beta}$-representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of $\boldsymbol{\beta}$-representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry's theorem characterizing sequences of nonnegative integers that are the greedy $\boldsymbol{\beta}$-representations of some real number in the interval $[0,1)$. We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the $\boldsymbol{\beta}$-shift is sofic if and only if all quasi-greedy $\boldsymbol{\beta}^{(i)}$-expansions of $1$ are ultimately periodic, where $\boldsymbol{\beta}^{(i)}$ is the $i$-th shift of the Cantor real base $\boldsymbol{\beta}$.