Take an irrational number ξ and form the sequence ξ(−b)ⁿ modulo 1 — the fractional parts of an alternating geometric progression. The signs alternate, so the sequence bounces between regions of [0,1) rather than spiraling in one direction. The question: how tightly can this sequence be confined?
The paper on alternating geometric progressions and Sturmian words (arXiv: 2603.22045) characterizes all ξ for which the sequence stays inside an interval of length b⁻¹ + b⁻² − b⁻³.
The characterization uses Sturmian words — infinite binary sequences that encode irrational rotations on the circle. A Sturmian word with angle α has exactly n+1 distinct factors of length n, making it the simplest possible aperiodic sequence. The connection: the alternating geometric progression mod 1 traces a pattern that is Sturmian when the confinement is tight.
The length threshold b⁻¹ + b⁻² − b⁻³ is sharp — no shorter interval can contain any such sequence (proved in prior work). The surviving ξ are those whose continued fraction expansion has a specific structure dictated by the Sturmian combinatorics.
The through-claim: the tightest confinement of an alternating sequence is governed by the simplest aperiodic structure. Sturmian words — the boundary between periodic and chaotic — characterize the boundary between confinement and escape. The irrational numbers that produce maximally confined sequences are those whose arithmetic (continued fractions) aligns with the combinatorics (Sturmian factors).
2603.22045. Number theory / Sturmian words / equidistribution / geometric progressions / continued fractions.