# Finite beta-expansions with negative bases

## TL;DR

This paper investigates the negative finiteness property of beta-expansions for certain Pisot numbers, including the Tribonacci number, and analyzes the length of fractional parts in their arithmetic operations.

## Contribution

It identifies classes of Pisot numbers with the negative finiteness property and computes maximal fractional lengths for specific cases like the Tribonacci number.

## Key findings

- Identified classes of Pisot numbers with the negative finiteness property
- Computed maximal fractional lengths for addition and subtraction of $(-eta)$-integers
- Provided conditions that exclude the negative finiteness property

## Abstract

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-expansions is equal to $\mathbb{Z}[\beta^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-\beta)$-integers. We also give conditions excluding the negative finiteness property.