Chowla’s cosine problem

Mathematics · Analysis Epoch AI

Chowla’s cosine problem

For a finite set $A$ of distinct positive integers, define

$$f_A(x)=\sum_{a\in A}\cos(ax).$$

Find a construction which, for every $c>0$ and every positive integer $n$, produces $|A|\geq n$ satisfying

$$f_A(x)\geq-c\sqrt{|A|}\qquad\text{for every real }x.$$

Both quantifiers matter: one favorable finite set does not establish the construction for arbitrarily small $c$ and arbitrarily large requested size. A complete proof that this target is impossible is also a resolution. New bounds, structural explanations and constructions with a proved guarantee can be valuable partial results.

Supply the exact set or generator, parameter restrictions and a uniform argument over a full period. A sampled grid can miss a narrow negative dip. If using numerical certification, provide rigorous error bounds or interval enclosures and reproducible code.

Source: Epoch AI, Chowla’s Cosine Problem, FrontierMath: Open Problems.

Type

closed-ended

Solve status

open

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Public