Szemeredi定理的Hardy域扩张
A Hardy field extension of Szemeredi's Theorem
摘要：
在1975年Szemerédi证明了一组整数的积极上密度包含任意长的算术级数。Bergelson和Leibman在1996年证明了等差数列的公差可以是正方形、立方体，或者更一般的形式为$p(n)$，其中$p(n)$是任何常数项为零的整数多项式。我们产生了各种新的结果，这种类型的序列，不是多项式。证明了Szemerédi定理中级数的公差可以是$[n^δ]$的形式，其中$δ$是任意正实数，$[x]$表示$x$的整数部分.更一般地，公差可以是$[a(n)]$的形式，其中$a(x)$是作为哈代域的成员并且对于某个非负整数$k$满足$a(x)/x^k\to \infty$和$a(x)/x^{k+1}\to 0$的任何函数。证明结合了一个新的结构的Hardy序列的结果，遍历理论的技术，和一些最近的序列的均匀分布的结果。
英文摘要：
In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form $p(n)$ where $p(n)$ is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi's theorem can be of the form $[n^δ]$ where $δ$ is any positive real number and $[x]$ denotes the integer part of $x$. More generally, the common difference can be of the form $[a(n)]$ where $a(x)$ is any function that is a member of a Hardy field and satisfies $a(x)/x^k\to \infty$ and $a(x)/x^{k+1}\to 0$ for some non-negative integer $k$. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds.
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