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arXiv · 2011.12275

Simultaneous small fractional parts of polynomials

James Maynard
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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 $c/k$.

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James Maynard. 2020-11-24. Simultaneous small fractional parts of polynomials. https://arxiv.org/abs/2011.12275

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