On the uniform distribution in residue classes of dense sets of integers with distinct sums

Abstract

A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this size we show that, under mild assumptions on the size of the modulus $m$ and on the difference $N^{1/2}-\Abs{{\cal A}}$ (these quantities should not be too large) the elements of ${\cal A}$ are uniformly distributed in the residue classes mod $m$. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lindstr\"om whose approach was combinatorial. Our main tool is an upper bound on the minimum of a cosine sum of $k$ terms, $\sum_1^k \cos{\lambda_j x}$, all of whose positive integer frequencies $\lambda_j$ are at most $(2-\epsilon)k$ in size.