Arithmetics in number systems with cubic base

TL;DR

This paper investigates greedy number representations in cubic base systems, analyzing how addition and multiplication affect digit expansions, especially for cubic Pisot units, and establishing bounds on extra digits that appear during these operations.

Contribution

It provides new bounds on the number of additional digits generated during arithmetic operations in cubic base systems, focusing on cubic Pisot units, and explores their implications.

Findings

Bounds on extra digits during addition and multiplication in cubic bases

Special focus on cubic Pisot units and their properties

Insights into the structure of greedy expansions in these systems

Abstract

This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits assuming the finiteness of the expansion of the considered sum or product, especially for the case of cubic Pisot units.