Simultaneous small fractional parts of polynomials

TL;DR

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

Contribution

It establishes an optimal bound on simultaneous small fractional parts of polynomials, refining earlier results by Schmidt.

Findings

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

Improved bound from c/k^2 to c/k in fractional parts estimate

Results are essentially optimal in the number of polynomials k

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 integer $n<x$ such that the fractional parts $\|f_i(n)\|\ll x^{c/k}$ for all $1\le i\le k$ and for some constant $c=c(d)$ depending only on $d$. This is essentially optimal in the $k$-aspect, and improves on earlier results of Schmidt who showed the same result with $c/k^2$ in place of $c/k$.