An improved upper bound for the size of the multiplicative 3-Sidon sets

Abstract

We say that a set is a multiplicative 3-Sidon set if the equation $s_1{s}_2{s}_3=t_1{t}_2{t}_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots ,n\}$ is at most $\pi (n)+\pi (n/2)+n^{2/3}(\log n{)}^{2^{1/3}-1/3+o(1)}$, which improves the previously known best bound $\pi (n)+\pi (n/2)+c{n}^{2/3}\log n/\log \log n$.