Parry condition, existence and uniqueness of alternate bases

TL;DR

This paper investigates the conditions under which a given expansion can serve as the base expansion of 1 in alternate numeration systems, extending known results from the Re9nyi case and exploring existence and partial uniqueness of such bases.

Contribution

It establishes new conditions for the existence of bases with specified expansions of 1 in alternate systems and introduces a fixed point approach to determine these bases.

Findings

Derived conditions for expansions of 1 in alternate bases.

Proved existence of bases with given sequences of B-integers.

Provided partial results on the uniqueness of these bases.

Abstract

Alternate bases are a numeration system that generalizes the R\'enyi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the R\'enyi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers.