On small fractional parts of polynomials

Abstract

We prove that for any real polynomial $f(x)\in \mathbb{R}[x]$ the set $$\{\alpha \in \mathbb{R}:\underset{n\to \infty }{\mathrm{liminf}}n\log n||\alpha f(n)||>0\}$$ 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.

8 pages