PreprintCC0arXiv
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 3m. First, for every fixed period p, the number of positions at which the binary expansion of 3m breaks p-periodicity grows in order like log m/loglog 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 3m has full low-order subword complexity: its complexity function satisfies p3m(n)≥ n+1 for every length n, once m is large enough.
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BibTeX
@misc{stephan2026aperiodicity,
title = {{Aperiodicity and subword complexity in the binary expansion of powers of three}},
author = {Ralf Stephan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.14774},
eprint = {2607.14774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14774},
note = {CC0},
}
RIS
TY - UNPB
TI - Aperiodicity and subword complexity in the binary expansion of powers of three
AU - Ralf Stephan
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2607.14774
UR - https://arxiv.org/abs/2607.14774
AB - 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/loglog 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)≥ n+1 for every length n, once m is large enough.
ER -