On small fractional parts of polynomials

We prove that for any real polynomial $f(x) \in\mathbb{R} [x]$ the set $$\{\alpha \in \mathbb{R}: \liminf_{n\to \infty} n\log n ||\alpha f(n)|| >0\}$$ has positive Hausdorff dimension. Here $||\xi ||$ means the distance from $\xi$ to the nearest integer. Our result is based on an original method due to Y. Peres and W. Schlag.