Divisibility properties of polynomial expressions of random integers

Abstract

We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random from the set $\{1,\ldots,n\}^s$, $s\in\mathbb{N}$. We show that the ${\rm GCD}$ and the suitably normalized ${\rm LCM}$ of this set converge in distribution to a.s.\ finite random variables under mild assumptions on $f_1,\ldots, f_m$. Our approach is based on the notion of integer adeles and a known fact that the uniform distribution on $\{1,\ldots, n\}$ converges to the Haar measure on the ring of integer adeles combined with the Lang-Weil bounds.