An upper bound on the size of Sidon sets

Abstract: In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of ${ 1, 2, \ldots, n}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.