Chowla's cosine problem

Definitions and mathematical statement

For a nonempty finite set $A\subset\{1,2,\ldots\}$ of distinct positive integer frequencies, define the real trigonometric polynomial \[ C_A(x)=\sum_{a\in A}\cos(ax),\qquad m(A)=-\min_{x\in[0,2\pi]}C_A(x). \] Continuity and periodicity ensure that this inner minimum exists. For $n\ge1$, put \[ m(n)=\inf\{m(A):A\subset\mathbb N,\ |A|=n\}. \] The source writes an outer minimum; the infimum convention defines the extremal value without presupposing attainment. The problem is to determine this value, or its sharp order of growth as $n\to\infty$, and to determine attainment if one uses the source's minimum notation. An order-of-growth answer means matching upper and lower bounds with absolute positive multiplicative constants, uniform in $n$.

Every coefficient is exactly one; repetitions of frequencies and a constant term are excluded. There is no upper bound on the elements of $A$. Since $C_A$ has average zero and $C_A(0)=n>0$, its minimum is negative. Thus $m(A)$ measures the unavoidable negative excursion, and the optimization asks how small that excursion can be made by choosing the integer frequencies.

Short English statement

How small can the largest negative excursion of a sum of n distinct integer-frequency cosines be?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3346 (OPG-739), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Chowla's cosine problem. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/chowlas_cosine_problem.