On the greatest common divisor of integer parts of polynomials

TL;DR

This paper derives asymptotic formulas for counting tuples of positive integers and polynomial parts that are coprime, with error terms influenced by the Diophantine nature of polynomial coefficients.

Contribution

It provides new asymptotic formulas for coprime tuples involving polynomial parts, addressing a question posed by Bergelson and Richter.

Findings

Asymptotic formulas for coprime tuples involving polynomial parts

Error terms depend on Diophantine properties of polynomial coefficients

Addresses a question by Bergelson and Richter

Abstract

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error terms in our formulas are of various strengths depending on the Diophantine properties of the leading coefficients of these polynomials.