Aperiodicity and subword complexity in the binary expansion of powers of three

Abstract

We prove two results on the fine structure of the binary digits of $3^m$. First, for every fixed period $p$, the number of positions at which the binary expansion of $3^m$ breaks $p$-periodicity grows in order like $\log m/\log \log m$; equivalently, no window of the expansion deeper than a fixed power of $\log m$ is $p$-periodic. Second, the finite binary word formed by the low-order digits of $3^m$ has full low-order subword complexity: its complexity function satisfies $p_{3^m}(n)\ge n+1$ for every length $n$, once $m$ is large enough.

Note added