Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Stephan, Ralf
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169558220