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Extremal cosine sign correlations
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 41−n and construct frequency sets for which this proportion is at most 16nθ4−n, where θ=log2(3/2). These bounds determine the optimal exponential decay rate 1/4.
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BibTeX
@misc{zang2026extremal,
title = {{Extremal cosine sign correlations}},
author = {Shengyang Zang and Yingyu Zhao and Yunhao Zhao and Chen Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33190},
eprint = {2610.33190},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33190},
}
RIS
TY - UNPB
TI - Extremal cosine sign correlations
AU - Shengyang Zang
AU - Yingyu Zhao
AU - Yunhao Zhao
AU - Chen Zhou
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33190
UR - https://arxiv.org/abs/2610.33190
AB - 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^θ4^−n, where θ=log_2(3/2). These bounds determine the optimal exponential decay rate 1/4.
ER -