On the probability that the integer parts of polynomials are coprime

Abstract

We prove that for any $s\ge 2$ and polynomials $f_{1}(x),\dots,f_{s-1}(x)\in \mathbb{R}[x]$ satisfying a certain equidistribution condition, the natural density of the numbers $n\in \mathbb{N}$ such that $n,\lfloor f_{1}(n)\rfloor,\dots,\lfloor f_{s-1}(n)\rfloor$ are mutually coprime is $1/ζ(s)$. This generalises a recent result of Champagne, in which this was proven under the assumption that $f_{1}$ is linear.