Sidon sets in a union of intervals (Q2095130)

A Sidon set \(A\) is a subset of \(\mathbb{N}\) with the property that all sums of two elements are distinct. It is well-known that the maximum size of a Sidon set in an interval of size \(n\) is asymptotically equivalent to \(\sqrt{n}\). In this paper, a lower bound is given for the maximum size of a Sidon set in a union of two intervals. Conversely, by using the Erdős--Turán small difference technique, an upper bound is established for the maximum cardinality of Sidon sets in a union of \(k\) intervals. Namely, if \(A\) is the union of two intervals of respectively size \(n_1\) and \(n_2\), the maximum cardinality of a Sidon set in \(A\) is asymptotically between \(0,876\sqrt{n_1+n_2}\) and \(\sqrt{n_1+n_2}\).