On small fractional parts of perturbed polynomials

TL;DR

This paper investigates small fractional parts of perturbed polynomials, providing improved bounds on how closely fractional parts of polynomial evaluations at primes can approximate integers, extending previous results in analytic number theory.

Contribution

It establishes new bounds for fractional parts of perturbed polynomials evaluated at primes, with explicit dependence on polynomial degree and perturbation shape.

Findings

Improved bounds on fractional parts for perturbed polynomials.

Explicit dependence of bounds on polynomial degree and shape.

Extension of previous work by Madritsch and Tichy.

Abstract

Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on earlier work by Madritsch and Tichy. In particular, let $f=P+\phi$ where $P$ is a polynomial of degree $k$ and $\phi$ is a linear combination of functions of shape $x^c$, $c\not \in \mathbb{N}$, $1<c<k$. We prove that for any given irrational $\xi$ we have \[\min_{\substack{2\leq p\leq X\\ p \text{ prime}}} \Vert \xi \lfloor f(p)\rfloor\Vert \ll_{f,\epsilon} X^{-\rho(k)+\epsilon},\] for $P$ belonging to a certain class of polynomials and with $\rho(k)>0$ being an explicitly given rational function in $k$.