Alternating geometric progressions modulo one and Sturmian words

Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $\xi$ such that the image in $\mathbb{R}\mathrm{/}\mathbb{Z}$ 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.