Ruzsa’s Problem on Bi-Sidon Sets

Abstract

A subset S of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of S are distinct. Imre Ruzsa asked the following question: What is the maximum number f(N) such that every set S of N real numbers contains a bi-Sidon subset of size at least f(N)? He proved that \(f(N)\geqslant cN^{\frac{1}{3}}\), for a constant \(c>0\). In this note, we improve this bound to \(N^{\frac{1}{3}+\frac{7}{78}+o(1)}\).