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

TL;DR

This paper establishes a tighter upper bound on the size of multiplicative 3-Sidon sets within the first n natural numbers, advancing understanding of their combinatorial limitations.

Contribution

It provides an improved upper bound for the maximum size of multiplicative 3-Sidon sets, refining previous estimates with a more precise asymptotic expression.

Findings

New upper bound improves previous results

Bound involves prime counting functions and logarithmic factors

Advances theoretical understanding of multiplicative Sidon sets

Abstract

We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ 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 $\pi(n)+\pi(n/2)+cn^{2/3}\log n/\log\log n$.