Simultaneously Small Fractional Parts of Polynomials

Abstract

Let $f_1,\dots ,f_k\in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\cdots =f_k(0)=0$. We show that there is an $n<x$ such that $\Vert f_i(n)\Vert \ll x^{-1/10.5kd(d-1)+o(1)}$ 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 $d$.

19 pages. arXiv admin note: text overlap with arXiv:2011.12275 by other authors