论文标题
多项式的同时小部分部分
Simultaneous small fractional parts of polynomials
论文作者
论文摘要
令$ f_1，\ dots，f_k \ in \ mathbb {r} [x] $是$ d $的多项式，$ f_1（0）= \ dots = f_k（0）= 0 $。我们表明，有一个整数$ n <x $，因此所有$ 1 \ le i \ le i \ le k $ to the零件$ \ | f_i（n）\ | \ ll x^{c/k} $，对于某些常数$ c = c（d）$，仅根据$ d $。这在$ k $的角度实质上是最佳的，并且在施密特的早期结果中提高了$ c/k^2 $的结果，并取得了$ c/k $的结果。
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 $c/k$.
