P-交多项式的Sarkozy定理
Sarkozy's Theorem for P-Intersective Polynomials
摘要：
本文定义了多项式$h\in \mathbb{Z}[x]$的一个充要条件，以保证每一个正上密度的自然数集合对于某个素数$p$都包含$h(p)$形式的非零差.此外，我们建立了一个定量估计的大小最大的子集${1,2,\dots,N}$缺乏所需的算术结构，表明如果度$(h)=k$，那么这样一个集合的密度是最多一个常数倍$(\log N)^{-c}$的任何$c<1/(2k-2)$。我们还讨论了如何改进版本的$k=2$和一个相对版本的素数可以得到与一些额外的已知的方法。
英文摘要：
We define a necessary and sufficient condition on a polynomial $h\in \mathbb{Z}[x]$ to guarantee that every set of natural numbers of positive upper density contains a nonzero difference of the form $h(p)$ for some prime $p$. Moreover, we establish a quantitative estimate on the size of the largest subset of ${1,2,\dots,N}$ which lacks the desired arithmetic structure, showing that if deg$(h)=k$, then the density of such a set is at most a constant times $(\log N)^{-c}$ for any $c<1/(2k-2)$. We also discuss how an improved version of this result for $k=2$ and a relative version in the primes can be obtained with some additional known methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
