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Exact Minima of Finite Absolute Cosine Sums
Abstract
For a positive integer n, let Mn=minx∈ℝ∑k=1n |cos(kx)|. We determine Mn exactly. Apart from the exceptional values M2=1/√2, M4=1+√3/2, and M6=(−1+3√5+2√(5+2√5))/4, one has Mn=⌊ n/2⌋. The minimizers are also classified: for n∉{2,4,6}, equality is attained only at x≡ π/2pmodπ, while the exceptional cases are attained at x≡±π/4, ±π/6, and ±π/10pmodπ, respectively. The proof is elementary and uses piecewise concavity, a permutation modulo 2q, and two finite trigonometric estimates.
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BibTeX
@misc{sun2026exact,
title = {{Exact Minima of Finite Absolute Cosine Sums}},
author = {Ruixi Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05545},
eprint = {2609.05545},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05545},
}
RIS
TY - UNPB
TI - Exact Minima of Finite Absolute Cosine Sums
AU - Ruixi Sun
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.05545
UR - https://arxiv.org/abs/2609.05545
AB - For a positive integer n, let M_n=min_x∈ℝ∑_k=1^n |cos(kx)|. We determine M_n exactly. Apart from the exceptional values M_2=1/√2, M_4=1+√3/2, and M_6=(−1+3√5+2√(5+2√5))/4, one has M_n=⌊ n/2⌋. The minimizers are also classified: for n∉{2,4,6}, equality is attained only at x≡ π/2pmodπ, while the exceptional cases are attained at x≡±π/4, ±π/6, and ±π/10pmodπ, respectively. The proof is elementary and uses piecewise concavity, a permutation modulo 2q, and two finite trigonometric estimates.
ER -