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

Abstract

Write $(3/2)^n = m_n + \eps_n$ with $m_n$ the nearest integer and $\eps_n\in[-\tfrac12,\tfrac12)$, 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. Using results by Corvaja--Zannier and Nair--Kumar--Rout we prove that the subword complexity $\pT(k)$ of $T$ is superlinear, $\pT(k)/k\to\infty$. The argument is completely formalized in Lean-4, depending only on the Subspace Theorem.