Beta Expansions for Regular Pisot Numbers

TL;DR

This paper provides a complete classification of greedy beta expansions of 1 for all regular Pisot bases less than 2, revealing patterns and confirming a conjecture about cyclotomic co-factors.

Contribution

It explicitly determines the expansions for all regular Pisot numbers less than 2 and addresses Boyd's conjecture on cyclotomic co-factors.

Findings

Complete list of expansions for all regular Pisot bases less than 2

Identification of remarkable patterns in expansions

Verification of Boyd's conjecture on cyclotomic co-factors

Abstract

A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is known that if the base is a Pisot number, then this expansion will always be finite or periodic. Some work has been done to learn more about these expansions, but in general these expansions were not explicitly known. In this paper, we present a complete list of the greedy beta expansions of 1 where the base is any regular Pisot number less than 2, revealing a variety of remarkable patterns. We also answer a conjecture of Boyd's regarding cyclotomic co-factors for greedy expansions.