Sumsets of Sidon sets

1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence ~N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that ${\displaystyle N^{\frac{1}{3}}}$ is possible. In [1] they show that A + A contains gaps longer than clogN, while the maximal gap may be of size O(√N). We improve these bounds. In Section 2, we give an example of A + A containing an interval of length c√N; hence in this question the answer is known up to a constant factor. In Section 3, we construct A such that the maximal gap is ${\displaystyle \ll N^{\frac{1}{3}}}$. In Section 4, we construct A such that the maximal gap of A + A is O(logN) in a subinterval of length cN.