论文标题
Paley图和Sárközy的功能领域定理
Paley Graphs and Sárközy's Theorem In Function Fields
论文作者
论文摘要
Sárközy的定理指出，密集的整数集必须包含两个元素，其差异为$ k^{th} $ power。遵循CROOT，LEV和PACH的多项式方法的突破，Green证明了此结果的强大定量版本，对于$ \ Mathbb {f} _ {q} [t] $。在本文中，我们通过在$ \ mathbb {z} $中调整Ruzsa的构造来为Sárközy的函数字段定理提供了一个下限。我们构建了$ a $ a $ dem $ <n $的$ a $ a $不包含$ k^{th} $功率差的$ | a | = q^{n-n-n/2k} $。此外，我们证明了有关广义Paley图的独立性数量的少数结果，包括对Ruzsa主张的概括，这有助于理解该方法的极限。
Sárközy's theorem states that dense sets of integers must contain two elements whose difference is a $k^{th}$ power. Following the polynomial method breakthrough of Croot, Lev, and Pach, Green proved a strong quantitative version of this result for $\mathbb{F}_{q}[T]$. In this paper we provide a lower bound for Sárközy's theorem in function fields by adapting Ruzsa's construction for the analogous problem in $\mathbb{Z}$. We construct a set $A$ of polynomials of degree $<n$ such that $A$ does not contain a $k^{th}$ power difference with $|A|=q^{n-n/2k}$. Additionally, we prove a handful of results concerning the independence number of generalized Paley Graphs, including a generalization of a claim of Ruzsa, which helps with understanding the limit of the method.
