On small fractional parts of polynomials

Details On small fractional parts of polynomials On small fractional parts of polynomials

2007-11-12

arxiv.org/abs/0711.1753v1

Mathematics

Number Theory

8 pages

Scientific paper

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.

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