The number of integer points close to a polynomial

TL;DR

This paper establishes upper bounds on the number of integer solutions within a specified interval where a polynomial's value is close to an integer, advancing understanding of Diophantine approximation for polynomials.

Contribution

It provides new upper bounds for the count of integer points near a polynomial on a large interval, extending previous results in Diophantine approximation.

Findings

Derived upper bounds for solutions to (x) (x) (x) (x) (x) (x) (x) (x) (x) (x) (x) (x) (x) (x) (x) (x) (x)

Extended bounds to polynomial degrees n   (x) (x) (x) (x)

Abstract

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the inequality $\|f(x)\| \le \delta$ with $x \in [X,2X] \cap \mathbb{N}$.