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