Chowla's cosine problem

Importance: High ✭✭✭
Author(s): Chowla, Sarvadaman
Subject: Number Theory
Recomm. for undergrads: no
Posted by: mdevos
on: February 22nd, 2008

\begin{problem} Let $A \subseteq {\mathbb N}$ be a set of $n$ positive integers and set \[m(A) = - \min_x \sum_{a \in A} \cos(ax).\] What is $m(n) = \min_A m(A)$? \end{problem}

It is easy to see that $m(A) > 0$, since the average value of the sum of the cosines is zero. Bourgain [B] proved that $m(n) > e^{(\log n)^c}$ for some $c>0$ and $n$ sufficiently large. Recently, Ruzsa [R] tightened this argument, proving that $m(n) > c_1 e^{c_2 \sqrt{ \log n}}$ where $c_2 = \sqrt{ (\log 2)/ 8}$. The proof utilizes a clever manipulation of norms to reveal a (somewhat surprising) additive structure to the problem.

It seems the only known upper bound is $m(n) \ll \sqrt{n}$.


[B] J. Bourgain, Sur le minimum d'une somme de cosinus, Acta Arith. 45 (1986), 381--389. \MRhref{MR0847298}

*[C] S. Chowla, Some applications of a method of A. Selberg. J. Reine Angew. Math. 217 (1965) 128--132. \MRhref{MR0172853}

[R] I.Z. Ruzsa, Negative values of cosine sums. Acta Arith. 111 (2004), no. 2, 179--186. \MRhref{MR2039421}

* indicates original appearance(s) of problem.