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Superlinear complexity of the (3/2)n(3/2)^n steering word

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Contributed by Ralf Stephan ↗

Version 1 / Oct 20, 2026/ arXiv: 2607.11648 / CC0 1.0

Abstract

Write (3/2)n=mn+εn(3/2)^n = m_n + \varepsilon_n with mnm_n the nearest integer and εn∈[−12,12)\varepsilon_n\in[-\tfrac12,\tfrac12), and let T=(tn)T=(t_n), tn=2mn+1−3mnt_n=2m_{n+1}-3m_n, be the resulting \emph{steering word}: the step-by-step record of the map x↦32xx\mapsto\tfrac32 x on the orbit of 1, coded by nearest-integer rounding. We prove that the subword complexity pT(k)p_T(k) of TT is superlinear, pT(k)/k→∞p_T(k)/k\to\infty. A pigeonhole argument reduces this to exponential pair repulsion of the orbit modulo one, which we establish by two applications of the pp-adic Subspace Theorem: once in three variables, which handles all gaps between the two orbit points at once, and once in two variables (Ridout's theorem) for the single degenerate family of constant gap. The argument is completely formalized in Lean~4, together with the Subspace Theorem it uses.

Provenance statement

Author directed the LLM (Fable 5) at finding new results in the uniform distribution mod one of (3/2)^n problem space. Author directed Opus 4.8 and 5 to formalize Fable's plan, and Fable to write the draft paper, which underwent several versions. Yann Bugeaud read and commented on the paper; he also suggested the existence of a simplified proof, which Fable 5.1 subsequently found. Author directed LLM to adapt the paper accordingly.

Formalizations

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Version history

  1. v1 · 20 pagesSubmitted by Ralf StephanInitial depositCurrentOct 20, 2026