AI 中文总结
研究三的幂次二进制展开的精细结构，证明对于固定周期\(p\)，\(3^{m}\)二进制展开打破\(p\) - 周期性的位置数增长阶为\(\log m / \log\log m\)，且由\(3^{m}\)低阶数字形成的二进制字具有完整低阶子词复杂性。
AI 中文摘要
我们证明了关于\(3^{m}\)二进制数字精细结构的两个结果。首先，对于每个固定周期\(p\)，\(3^{m}\)二进制展开打破\(p\) - 周期性的位置数量增长阶为\(\log m / \log\log m\)；等价地，展开中深度超过\(\log m\)的固定幂次的窗口不是\(p\) - 周期性的。其次，由\(3^{m}\)低阶数字形成的有限二进制字具有完整的低阶子词复杂性：一旦\(m\)足够大，其复杂性函数对于每个长度\(n\)都满足\(\pcx_{3^{m}}(n) \geq n + 1\)。
英文摘要
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.
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