Gaps in Multiplicative Sidon Sets

Abstract

For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\subseteq\{1,2,\dots,n\}$ that intersects every interval $[x,x+L]\subseteq[1,n]$. S\'ark\"ozy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\le\sqrt n$ for every $n\in\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve the upper bound further and, with $\rho = \frac{13-\sqrt{69}}{10} < 0.47$, prove that $g(n)\ll_{\varepsilon} n^{\rho+\varepsilon}$ for every $\varepsilon > 0$.