On the Uniform Distribution in Residue Classes of Dense Sets of Integers with Distinct Sums Original Research Article

A set imagesubset of or equal to{1, …, N} is of the typeB2if all sumsa+b, withagreater-or-equal, slantedb,a, bset membership, variantimage, are distinct. It is well known that the largest such set is of size asymptotic toN1/2. For aB2set image of this size we show that, under mild assumptions on the size of the modulusmand on the differenceN1/2− image (these quantities should not be too large), the elements of image 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 Lindström whose approach was combinatorial. Our main tool is an upper bound on the minimum of a cosine sum ofkterms, ∑k1 cos λjx, all of whose positive integer frequenciesλjare at most (2−var epsilon) kin size.