Popular differences and generalized Sidon sets

Abstract

For a subset A⊆[N], we define the representation function rA−A(d):=#{(a,a′)∈A×A:d=a−a′} and define MD(A):=max1≤d<D⁡rA−A(d) for D>1. We study the smallest possible value of MD(A) as A ranges over all possible subsets of [N] with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of D. We shall also see how this problem connects to a well-known problem about generalized Sidon sets.