On the Uniform Distribution in Residue Classes of Dense Sets of Integers With Distinct Sums

A set A ` f1; : : : ; Ng is of type B 2 if all sums a+b, with a b, a; b 2 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 A of this size we show that, under mild assumptions on the size of the modulus m and on the difference N 1=2 \Gamma jAj (these quantities should not be too large) the elements of 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 Lindstrom whose approach was combinatorial. Our main tool is an upper bound on the minimum of a cosine sum of k terms, P k 1 cos j x, all of whose positive integer frequencies j are at most (2 \Gamma ffl)k in size. x0. Introduction and results A set A ` f1; : : : ; Ng is of type B 2 if all sums a + b; with a b; a; b 2 A; are distinct. (Such sets are also called Sidon, but the term has a very different meaning in harmonic analysis.) This is easily seen ..