Simultaneously Small Fractional Parts of Polynomials

TL;DR

This paper proves that for multiple polynomials with zero constant term, there exists an integer n less than x where all fractional parts are simultaneously very small, improving previous bounds especially regarding the degree of the polynomials.

Contribution

The authors establish a new bound on simultaneous small fractional parts of polynomials, extending prior work by incorporating the degree dependence into the exponent.

Findings

Existence of n < x with small fractional parts for all polynomials

Improved bound on the size of fractional parts considering polynomial degree

Enhanced understanding of fractional parts distribution for polynomial sequences

Abstract

Let $f_1,\dots,f_k \in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an $n<x$ such that $\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o(1)}$ for all $1\le i\le k$. This improves on an earlier result of Maynard, who obtained the same exponent dependency on $k$ but not on $d$.