Bedert 2025 polynomial bounds chowla cosine problem

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Benjamin Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).

For a finite set A of n positive integers and f_A(x) = sum_{a in A} cos(ax), Theorem 1.1 proves min_x f_A(x) <= -n^{1/5-o(1)}, hence K(n) >> n^{1/5-o(1)}, the first polynomial lower bound for Chowla's cosine problem; the previous record was Ruzsa's refinement K(n) >= e^{c'(log n)^{1/2}} of Bourgain's quasipolynomial bound K(n) > e^{(log n)^eps}. The paper first gives a short five-page proof of the weaker polynomial bound K(n) >> n^{1/12} (Section 5) and then improves the exponent to 1/5 with more work (Section 7); the Sidon-set construction gives K(n) << n^{1/2}, and Chowla conjectured K(n) of order n^{1/2}. The method builds on arithmetic input from Roth, Bourgain and Ruzsa (Section 4) plus preliminary observations about cosine polynomials (Section 3). Theorem 1.2, proved in Section 6, gives a polynomial one-sided bound for cosine polynomials whose coefficients lie in a fixed finite set S of nonzero reals; the author notes that all other methods giving superlogarithmic bounds for K(n) are very sensitive to the coefficients all lying in {0,1}. The author notes an independent contemporaneous preprint of Jin, Milojević, Tomon and Zhang reaching exponent 1/10-o(1) by a different, graph-eigenvalue route. For problem 510, Chowla's cosine problem, which asks for the growth of K(n), the paper gives the first polynomial lower bound, short of the conjectured exponent 1/2.

Source: https://arxiv.org/abs/2509.05260. The arXiv record (https://arxiv.org/abs/2509.05260, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #510

Results to transcribe.

• Theorem 1.1: Every set A ⊂ N with |A| = n satisfies min_{x in [0,2pi]} sum_{a in A} cos(ax) <= -n^{1/5-o(1)}, so K(n) >> n^{1/5-o(1)}.
• Section 5 bound: A streamlined five-page argument already yields the polynomial bound K(n) >> n^{1/12}.
• Theorem 1.2 (proved in Section 6): Let S, a subset of R \ {0}, be finite. There are constants c_S, c'S > 0 such that for every symmetric set A in Z \ {0} with |A| = n and all coefficients s_a in S with s_a = s{-a}, min_{x in R} sum_{a in A} s_a e(ax) <= -c'S n^{c_S}, where e(t) = e^{2 pi i t}. This is a polynomial bound for K_S(n), against the previous general bound K_S(n) >> (min{s in S} |s|) log n.