PreprintCC0arXiv
Superlinear complexity of the (3/2)n steering word
Abstract
Write (3/2)n = mn + εn with mn the nearest integer and εn∈[−1/2,1/2), and let T=(tn), tn=2mn+1−3mn, be the resulting \emph{steering word}: the step-by-step record of the map x↦3/2 x on the orbit of 1, coded by nearest-integer rounding. We prove that the subword complexity pT(k) of T is superlinear, pT(k)/k→∞. A pigeonhole argument reduces this to exponential pair repulsion of the orbit modulo one, which we establish by two applications of the p-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.
Related
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BibTeX
@misc{stephan2026superlinear,
title = {{Superlinear complexity of the $(3/2)^n$ steering word}},
author = {Ralf Stephan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.11648},
eprint = {2607.11648},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.11648},
note = {CC0},
}
RIS
TY - UNPB
TI - Superlinear complexity of the (3/2)^n steering word
AU - Ralf Stephan
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2607.11648
UR - https://arxiv.org/abs/2607.11648
AB - Write (3/2)^n = m_n + ε_n with m_n the nearest integer and ε_n∈[−1/2,1/2), and let T=(t_n), t_n=2m_n+1−3m_n, be the resulting \emph{steering word}: the step-by-step record of the map x↦3/2 x on the orbit of 1, coded by nearest-integer rounding. We prove that the subword complexity p_T(k) of T is superlinear, p_T(k)/k→∞. A pigeonhole argument reduces this to exponential pair repulsion of the orbit modulo one, which we establish by two applications of the p-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.
ER -