Alternating geometric progressions modulo one and Sturmian words

TL;DR

This paper characterizes irrational numbers whose alternating geometric progressions modulo one are confined to a specific interval, using Sturmian words, and establishes minimal interval length constraints.

Contribution

It provides a complete description of such irrationals and proves the minimal possible interval length for the sequence images.

Findings

Identifies all irrationals with sequences in a specific interval

Shows the minimal interval length is exactly $b^{-1}+b^{-2}-b^{-3}$

Uses Sturmian words to describe the sequence behavior

Abstract

Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $\xi$ such that the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(\xi (-b)^n)_{n\ge 0}$ 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.