Open Problem Garden

Importance: High ✭✭✭

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Keywords:	circle
	cosine polynomial

Recomm. for undergrads: no

Posted	by:	mdevos
	on:	February 22nd, 2008

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 $m(n) = \min_A m(A)$ ?

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.