论文标题
$ \ MATHCAL {CR} $中的丰富算术进展
Abundance of arithmetic progressions in $\mathcal{CR}$-sets
论文作者
论文摘要
H.Furstenberg和E.Glasner证明，对于任意的$ k \ in \ Mathbb {n} $，任何分段的串联整数都包含$ k $ term arithmetic的进步，并且此类进步的收集本身是$ \ \ \ \ \ \ \ \ \ niuts nortary and intirary and intrary severnary secrign intrary sextrary and intrary sextrary sergers。 Hindman，使用离散半群的Stone-čech紧凑型的代数。但是，他们为各种大型集合提供了丰富的体现。在\ cite {dhs}中，第一作者尼尔·欣德曼（Neil Hindman）和多娜·斯特劳斯（Dona Strauss）介绍了两个大型概念，即$ j $ set和$ c $ set。在\ cite {bg}中，V。Bergelson和D. Glasscock引入了另一个宽敞的概念，类似于$ J $ -SET的概念，即$ \ Mathcal {Cr} $ set。所有这些集合包含任意长度的算术进程。在\ cite {dg}中，第二作者和S. goswami证明，对于任何$ j $ - set，$ a \ subseteq \ mathbb {n} $，集合$ \ {（a，a，b）：\，\，\ {a，a+b，a+b，a+b，a+2b，a+ldots a $ lb \ y $ j $ y $ $（\ mathbb {n \ times \ mathbb {n}}，+）$。在本文中，对于$ \ Mathcal {Cr} $ - SETS，我们证明了这一点。
H.Furstenberg and E.Glasner proved that for an arbitrary $k\in\mathbb{N}$, any piecewise syndetic set of integers contains a $k$-term arithmetic progression and the collection of such progressions is itself piecewise syndetic in $\mathbb{Z}.$ The above result was extended for arbitrary semigroups by V. Bergelson and N. Hindman, using the algebra of the Stone-Čech compactification of discrete semigroups. However, they provided an abundance for various types of large sets. In \cite{DHS}, the first author, Neil Hindman and Dona Strauss introduced two notions of large sets, namely, $J$-set and $C$-set. In \cite{BG}, V. Bergelson and D. Glasscock introduced another notion of largeness, which is analogous to the notion of $J$-set, namely $\mathcal{CR}$- set. All these sets contain arithmetic progressions of arbitrary length. In \cite{DG}, the second author and S. Goswami proved that for any $J$-set, $A\subseteq\mathbb{N}$, the collection $\{(a,b):\,\{a,a+b,a+2b,\ldots,a+lb\}\subset A\}$ is a $J$-set in $(\mathbb{N\times\mathbb{N}},+)$. In this article, we prove the same for $\mathcal{CR}$-sets.
