The number of integer points close to a polynomial

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