On small fractional parts of polynomials

We prove that for any real polynomial $f(x)\in \mathbb{R}[x]$ the set $$\{\alpha \in \mathbb{R}:\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}n\log n\mathrm{\mid }\mathrm{\mid }\alpha f(n)\mathrm{\mid }\mathrm{\mid }>0\}$$ has positive Hausdorff dimension. Here $\mathrm{\mid }\mathrm{\mid }\xi \mathrm{\mid }\mathrm{\mid }$ means the distance from $\xi$ to the nearest integer. Our result is based on an original method due to Y. Peres and W. Schlag.