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