On strong Sidon sets of integers

A set S⊂N of positive integers is a Sidon set if the pairwise sums of its elements are all distinct, or, equivalently, if |(x+w)−(y+z)|≥1 for every x,y,z,w∈S with x<y≤z<w. Let 0≤α<1 be given. A set S⊂N is an α-strong Sidon set if |(x+w)−(y+z)|≥w^α for every x,y,z,w∈S with x<y≤z<w. We prove that the existence of dense strong Sidon sets implies that randomly generated, infinite sets of integers contain dense Sidon sets. We derive the existence of dense strong Sidon sets from Ruzsa's well known result on dense Sidon sets [J. Number Theory 68 (1998), no. 1, 63–71]. We also consider an analogous definition of strong Sidon sets for sets S contained in [n]={1,…,n}, and give good bounds for F(n,α)=max⁡|S|, where S ranges over all α-strong Sidon sets contained in [n].