Simultaneous Small Fractional Parts of Polynomials

Abstract

Let f_1,dots ,f_kin 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 Vert f_i(n)Vert _{mathbb {R}/mathbb {Z}}ll x^{c/k} for all 1le ile k 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.