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Greedy Beta-expansions for families of Salem numbers

Kevin G. Hare, Liam Orovec

TL;DR

The paper addresses greedy $eta$-expansions of $1$ for Salem bases that converge to a Pisot limit, linking these expansions to the Pisot-base case via companion polynomials and Parry’s criterion. It extends Hare–Tweedle’s positive-case results to periodic and certain nonfinite settings, providing explicit expansion patterns for infinite Salem families and establishing that, under reversibly greedy conditions, these patterns persist as the Salem index grows. The work derives both periodic and finite-expansion phenomena, proves companion-polynomial criteria that certify greediness, and supports the theory with extensive data on regular Pisot families. This advances understanding of how non-Pisot bases behave in beta-expansions, offering concrete templates for constructing greedy expansions in Salem families and suggesting wider applicability to regular Pisot structures and potential generalizations.

Abstract

We give criteria for finding the greedy $β$-expansion for $1$ for families of Salem numbers that approach a given Pisot number. We show that these expansions are related to the greedy expansion under the Pisot base. This expands on the work of Hare and Tweedle.

Greedy Beta-expansions for families of Salem numbers

TL;DR

The paper addresses greedy -expansions of for Salem bases that converge to a Pisot limit, linking these expansions to the Pisot-base case via companion polynomials and Parry’s criterion. It extends Hare–Tweedle’s positive-case results to periodic and certain nonfinite settings, providing explicit expansion patterns for infinite Salem families and establishing that, under reversibly greedy conditions, these patterns persist as the Salem index grows. The work derives both periodic and finite-expansion phenomena, proves companion-polynomial criteria that certify greediness, and supports the theory with extensive data on regular Pisot families. This advances understanding of how non-Pisot bases behave in beta-expansions, offering concrete templates for constructing greedy expansions in Salem families and suggesting wider applicability to regular Pisot structures and potential generalizations.

Abstract

We give criteria for finding the greedy -expansion for for families of Salem numbers that approach a given Pisot number. We show that these expansions are related to the greedy expansion under the Pisot base. This expands on the work of Hare and Tweedle.

Paper Structure

This paper contains 13 sections, 10 theorems, 55 equations, 7 tables.

Table of Contents

  1. Introduction
  2. The Positive Case
  3. Finite Case
  4. Periodic Case
  5. The Negative Case
  6. Periodic Case
  7. Finite Case
  8. Examples and Motivation
  9. Structure of Pisot Numbers
  10. Positive Periodic Examples
  11. Negative Periodic Examples
  12. Data
  13. Further Remarks

Key Result

Theorem 1.9

Let $\mathbf{a}=(a_n)_{n\geq 1}$ be a sequence in $\{0,1\}^{\mathbb{N}}$. That is, $\mathbf{a}$ is a non-empty finite or infinite sequence $a_1\dots a_k$ or $a_1a_2\dots$ with $a_i\in\{0,1\}$. Then the sequence $\mathbf{a}$ is the greedy expansion of $1$ for some $\beta>1$ if and only if for all $j\ Here $\sigma(a_1a_2\dots)=a_2a_3\dots$.

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9: Parry parry
  • Definition 1.10
  • ...and 29 more