On small fractional parts of polynomial-like functions

TL;DR

This paper improves bounds on the fractional parts of polynomial-like functions, specifically for functions where the non-integer exponent exceeds the polynomial degree and is greater than 4, refining previous Diophantine inequality results.

Contribution

It advances the understanding of fractional parts of polynomial-like functions by providing sharper bounds when the non-integer exponent exceeds the polynomial degree and is greater than 4.

Findings

Improved bounds for fractional parts when c>k and c>4

Enhanced Diophantine inequality results for polynomial-like functions

Refined estimates for fractional parts of functions with non-integer exponents

Abstract

In a recent paper, Madritsch and Tichy established Diophantine inequalities for the fractional parts of polynomial-like functions. In particular, for $f(x)=x^k+x^c$ where $k$ is a positive integer and $c>1$ is a non-integer, and any fixed $\xi\in [0,1]$ they obtained \[\min_{2\leq p\leq X} \Vert \xi \lfloor f(p)\rfloor \Vert\ll_{k,c,\epsilon} X^{-\rho_1(c,k)+\epsilon}\] for $\rho_1(c,k)>0$ explicitly given. In the present note, we improve upon their results in the case $c>k$ and $c>4$.