  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 $x \in [X,2X] \cap \mathbb{N}$.
