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.

Disclosure

“APERIODICITY AND SUBWORD COMPLEXITY OF 3m IN BINARY 19 6. Acknowledgements The author utilized Claude Code as an AI coding assistant to aid in the Lean 4 formalization of the proofs presented in this paper. The author directed and reviewed all generated code and takes full responsibility for the mathematical integrity and final content of”

PDF page 19

Classification

Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-23T18:55:58.156436Z

Structural counts