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arXiv · 2610.33190

Extremal cosine sign correlations

Shengyang Zang·Yingyu Zhao·Yunhao Zhao·Chen Zhou
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Abstract

We prove that four cosines with distinct positive integer frequencies have the same sign on at least $1/33$ of a period, with equality precisely for positive integer dilates of $\{1,3,11,33\}$. If both odd and even frequencies occur, the optimal proportion is $1/24$; equality holds precisely for odd positive integer dilates of $\{1,3,4,12\}$ and $\{2,4,5,15\}$. For arbitrary $n$, we establish the universal lower bound $4^{1-n}$ and construct frequency sets for which this proportion is at most $16n^\theta4^{-n}$, where $θ=\log_2(3/2)$. These bounds determine the optimal exponential decay rate $1/4$.

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Shengyang Zang, Yingyu Zhao, Yunhao Zhao, Chen Zhou. 2026-10-23. Extremal cosine sign correlations. https://arxiv.org/abs/2610.33190

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