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Alternating geometric progressions modulo one and Sturmian words

Qing Lu, Weizhe Zheng

Abstract

Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $ξ$ such that the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-b)^n)_{n\ge 0}$ is contained in an interval of length $b^{-1}+b^{-2}-b^{-3}$. In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval.

Alternating geometric progressions modulo one and Sturmian words

Abstract

Let be an integer. Using Sturmian words we describe all irrational real numbers such that the image in of the sequence is contained in an interval of length . In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval.
Paper Structure (4 sections, 21 theorems, 28 equations)

This paper contains 4 sections, 21 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The positive case
  4. Proofs of the main results

Key Result

Theorem 1.2

Let $b\ge 2$ be an integer and let $\xi$ be an irrational real number. Then the sequence $(\pi(\xi(-b)^n))_{n\ge 0}$ is contained in an interval of $\mathbb{R}/\mathbb{Z}$ of length $b^{-1}+b^{-2}-b^{-3}$ if and only if $\xi=g/(b+1)+t_{-1/b}(\mathbf{w})$, where $g$ is an integer and $\mathbf{w}\in \ where $\theta$ denotes the slope of $\mathbf{w}$, and the sequence is contained in a semiopen inter

Theorems & Definitions (42)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 32 more