Draft:Chowla's cosine problem

Chowla's cosine problem is a problem in harmonic analysis concerning the minimum of a sum of cosine functions with distinct positive integer frequencies. It asks how large a negative value such a sum must attain, in terms of the number of summands. The problem is named after Sarvadaman Chowla.

Statement

For a finite nonempty set $A\subset \mathbb {Z} _{>0}$ with $|A|=n$, define $$f_{A}(x)=\sum _{a\in A}\cos(ax).$$ The extremal quantity is $$K(n)=\inf _{A\subset \mathbb {Z} _{>0},\,|A|=n}\left(-\min _{x\in [0,2\pi ]}f_{A}(x)\right).$$ Although $f_{A}(0)=n$ gives its maximum, estimating its minimum uniformly over all such sets is more difficult. Chowla conjectured that $K(n)$ has order ${\sqrt {n}}$: there is an absolute constant $c>0$ such that every such set satisfies $$\min _{x\in [0,2\pi ]}f_{A}(x)\leq -c{\sqrt {n}}.$$ Constructions using Sidon sets give a matching upper bound $K(n)=O({\sqrt {n}})$.

History

The problem originated in a question raised by Nesmith Ankeny and Chowla in 1948 while studying Dedekind zeta functions. They asked whether the minimum must tend to negative infinity as the number of terms increases. Miyoko Uchiyama and Saburo Uchiyama answered this affirmatively. Chowla formulated the square-root conjecture in 1965.

The resolution of Littlewood's conjecture on the $L^{1}$ norm of exponential sums implied a logarithmic lower bound for $K(n)$. Jean Bourgain obtained a stronger bound, and Imre Z. Ruzsa refined his method to obtain a bound of the form $\exp(c{\sqrt {\log n}})$, with $c>0$.

Polynomial bounds

For six decades after Chowla's 1965 formulation, no general polynomial bound was known. In 2025, Benjamin Bedert and, independently, Zhihan Jin, Aleksa Milojević, István Tomon and Shengtong Zhang crossed this threshold. The latter group's October 2025 preprint gives $K(n)\geq n^{1/10-o(1)}$, using structural results in spectral graph theory. Their argument relates cosine sums to eigenvalues of Cayley graphs.

In the July 2026 revision of his preprint, Bedert obtained $$K(n)\geq n^{1/5-o(1)}.$$

In September 2026, Abhishek Shankar proved the improved bound $K(n)\geq cn^{1/5}$ for an absolute constant $c>0$, removing the logarithmic loss from Bedert's estimate. His proof combines two estimates from Bedert's work with a new exact counting identity for asymmetric boundaries of additive intersections. This identity replaces the multiplicative-amplification step, yielding a shorter argument and the full $1/5$-power bound. By averaging the boundary sizes, the proof identifies a sufficiently large boundary directly from the number of additive triples. Combining this conclusion with Bedert's boundary estimate gives a uniform bound valid for every finite nonempty set of distinct positive integer frequencies.

Computational work

Idris Mercer studied finite searches for extremal cosine sums, determining the extremal values for sums with two and three terms. His work also considers the difficulty of reducing the search over unbounded integer frequencies to a finite computation.

Further reading

• Cook, John D. (7 February 2026). "Minimum of cosine sum".