论文标题
Sárközy的定理在各种有限的现场设置中
Sárközy's Theorem in Various Finite Field Settings
论文作者
论文摘要
在本文中，我们在多项式环$ \ mathbb {f} _q [x] $的环境中加强了Green的结果。在整数设置中，对于给定的多项式$ f \ in \ mathbb {z} [x] $持续零，（概括）sarkozy seorem给出了一个最大尺寸的上限，最大尺寸$ = f（b）$ for \ in \ mathbb {z} $中的某些$ b \。格林被证明是一个类似的结果，在子集$ a \ subset \ mathbb {f} _q [x] $的子集的设置中的界限更加强，多项式环$ \ mathbb {f} _q [x] $的界限，但要求附加条件的其他条件，即polynomial $ f \ in \ mathb $ quile $ q $ q $ q $ q $ q $ q py $ q py $ q copr $ q copr。我们概括了格林的结果，消除了这种情况。作为一个应用程序，我们还获得了Sarkozy定理的版本，其subsets $ a \ subset \ mathbb {f} _q $ for $ q = p = p^n $的subsets $ a \ subset \ subset \ subset \ subset \ subset $ a \ subset定理的版本。
In this paper, we strengthen a result by Green about an analogue of Sarkozy's theorem in the setting of polynomial rings $\mathbb{F}_q[x]$. In the integer setting, for a given polynomial $F \in \mathbb{Z}[x]$ with constant term zero, (a generalization of) Sarkozy's theorem gives an upper bound on the maximum size of a subset $A \subset \{1, \ldots, n \}$ that does not contain distinct $a_1,a_2 \in A$ satisfying $a_1 - a_2 = F(b)$ for some $ b \in \mathbb{Z}$. Green proved an analogous result with much stronger bounds in the setting of subsets $A \subset \mathbb{F}_q[x]$ of the polynomial ring $\mathbb{F}_q[x]$, but required the additional condition that the number of roots of the polynomial $F \in \mathbb{F}_q[x]$ is coprime to $q$. We generalize Green's result, removing this condition. As an application, we also obtain a version of Sarkozy's theorem with similarly strong bounds for subsets $A \subset \mathbb{F}_q$ for $q = p^n$ for a fixed prime $p$ and large $n$.
