The number of integer points close to a polynomial

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 $\Vert \cdot \Vert$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the inequality $\Vert f(x)\Vert \le \delta$ with $x\in [X,2X]\cap \mathbb{N}$.

Communicated by Jean-Marie De Koninck, Received March 1, 2018; accepted April 18, 2018