Research Problems in Function Theory — Problem 4.19

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For distinct positive integers $n_1,\ldots,n_k$, let $S(x)=\sum_{j=1}^k\cos(n_jx)$ and let $\|S\|_1=\int_0^{2\pi}|S(x)|\,dx$. The frequencies vary with the summation index. This is the cosine-sum form identified in the original research source, which also gives its logarithmic lower-bound theorem. [S3]

Statement recorded in the supplied source.

Is there an absolute constant $c>0$ such that \[\int_0^{2\pi}\left|\sum_{j=1}^k\cos(n_jx)\right|\,dx>c\log k\] for every $k\ge1$ and every set of distinct positive frequencies? The historically stated conjecture was proved independently by Konyagin and by McGehee, Pigno and Smith, as recorded in the source update and the original research paper cited here. [S2][S3]

Short English statement

Does every sum of k cosine functions with distinct positive integer frequencies have integral absolute value bounded below by a fixed multiple of log k?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304019 (AMR-022-4019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.19, its complete Update 4.19, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.
• S3 Julian Sahasrabudhe, Counting Zeros of Cosine Polynomials: On a Problem of Littlewood, arXiv:1610.07680v3, 2019. Introduction, p. 2; Section 3, Theorem 8 and its immediate cosine-polynomial consequence, p. 8. https://arxiv.org/pdf/1610.07680v3.