On infinite multiplicative Sidon sets

TL;DR

This paper investigates the growth behavior of infinite multiplicative Sidon sets, establishing bounds on their size relative to prime counting and constructing examples with specific asymptotic properties.

Contribution

It proves bounds on the size difference between such sets and primes, and constructs a set with a positive lower limit of this difference.

Findings

Bounded the lim inf of size difference over a specific growth rate

Constructed a set with positive lim inf of size difference

Established asymptotic behavior of infinite multiplicative Sidon sets

Abstract

We prove that if $A$ is an infinite multiplicative Sidon set, then $\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}}<\infty$ and construct an infinite multiplicative Sidon set satisfying $\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}}>0$.