Parry condition, existence and uniqueness of alternate bases
This work addresses a theoretical gap in numeration systems for mathematicians, but it is incremental as it extends known results from the Rényi case to the alternate case.
The paper tackles the problem of determining when given words can be expansions of 1 in alternate bases, a generalization of the Rényi numeration system, by establishing conditions using a fixed point theorem on matrices and obtaining a partial uniqueness result.
Alternate bases are a numeration system that generalizes the Rényi 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 Rényi 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.