On small fractional parts of polynomials

TL;DR

This paper demonstrates that for any real polynomial, the set of real numbers with a certain fractional part property related to the polynomial's values has positive Hausdorff dimension, using a novel method.

Contribution

It introduces a new approach based on Peres and Schlag's method to analyze fractional parts of polynomial sequences and their Hausdorff dimension.

Findings

The set of real numbers with a positive lim inf of scaled fractional parts has positive Hausdorff dimension.

The result applies to any real polynomial, regardless of degree.

A new method for analyzing fractional parts of polynomial sequences is developed.

Abstract

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.