# Periodic representations in Salem bases

## TL;DR

This paper proves that all algebraic bases enable eventually periodic representations of elements in their number field with a finite digit set, and classifies bases with the weak greedy property, highlighting the complexity of related decision problems.

## Contribution

It establishes that algebraic bases support eventually periodic representations and classifies those with the weak greedy property, linking the problem to topological properties of attractors.

## Key findings

- All algebraic bases allow eventually periodic representations.
- Classification of bases with the weak greedy property is provided.
- Deciding such representations relates to complex topological properties.

## Abstract

We prove that all algebraic bases $\beta$ allow an eventually periodic representations of the elements of $\mathbb Q(\beta)$ with a finite alphabet of digits $\mathcal A$. Moreover, the classification of bases allowing that those representations have the so-called weak greedy property is given. The decision problem whether a given pair $(\beta,\mathcal A)$ allows eventually periodic representations proves to be rather hard, for it is equivalent to a topological property of the attractor of an iterated function system.