**Problem**Let be a set of positive integers and set What is ?

It is easy to see that , since the average value of the sum of the cosines is zero. Bourgain [B] proved that for some and sufficiently large. Recently, Ruzsa [R] tightened this argument, proving that where . 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 .

## Bibliography

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

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

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

* indicates original appearance(s) of problem.