An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases

TL;DR

This paper extends Solomyak's theorem to periodic Cantor real expansions in alternate bases, analyzing algebraic properties and Galois conjugates to deepen understanding of these numeration systems.

Contribution

It introduces an analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases, including bounds on Galois conjugates under algebraic assumptions.

Findings

Derived bounds for norms of Galois conjugates.

Extended Solomyak's results to new numeration systems.

Analyzed algebraic properties of bases with periodic expansions.

Abstract

In this paper, we consider the positional numeration system, called the Cantor real expansion, on the unit interval $[\gamma, \gamma+1]$, where $\gamma \in \mathbb{R}$, with respect to an alternate base (i.e., a base which is a purely periodic sequence of real numbers). In particular, we study the case where the expansion of $\gamma+1$ is periodic. Under certain assumptions, the base satisfies algebraic properties. We compute the bounds for the norms of the nontrivial Galois conjugates associated with the base; thereby, extending the results of Solomyak on the classical beta expansions.