{"id":"hexagon/2610.00414","pdf":"https://hexagonmath.org/pdf/2610.00414v1","url":"https://hexagonmath.org/abs/2610.00414v1","tags":["math.CO","msc:11J87","msc:68R15","msc:11K06","uniform distribution"],"score":0,"title":"Superlinear complexity of the $(3/2)^n$ steering word","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Ralf Stephan"],"reviews":0,"subject":"math.NT","summary":"Write $(3/2)^n = m_n + \\varepsilon_n$ with $m_n$ the nearest integer and $\\varepsilon_n\\in[-\\tfrac12,\\tfrac12)$, 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 $x\\mapsto\\tfrac32 x$ on the orbit of 1, coded by nearest-integer rounding. We prove that the subword complexity $p_T(k)$ of $T$ is superlinear, $p_T(k)/k\\to\\infty$. 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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry.html?id=PALOMAR-2026-10-20-000014&version=1"],"human_understanding":"all"}