A Quantitative Result on Diophantine Approximation for Intersective Polynomials

TL;DR

This paper extends the best known bounds for recurrence modulo 1 from squares to a broad class of intersective polynomials, advancing understanding in Diophantine approximation and polynomial recurrence.

Contribution

It generalizes Green and Tao's approach to a wider class of polynomials, providing new quantitative bounds in Diophantine approximation.

Findings

Extended recurrence bounds from squares to intersective polynomials

Discussed implications for polynomial structures in sumsets

Highlighted limitations of the current method

Abstract

In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomials structures in sumsets and limitations of the method.