In 1975, Szemeredi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemeredi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of limiting densities of sets along multidimensional arithmetic progressions. In this note we present an arrangement of this proof that incorporates a number of notational and technical simplifications. Firstly, we replace the use of the regularity lemma by that of the simpler ``weak regularity lemma'' of Frieze and Kannan. Secondly, we extract the key inductive steps at the core of Szemeredi's proof (referred to as ``Lemma 5'', ``Lemma 6'', and ``Fact 12'' in that paper) as stand-alone theorems that can be stated with less notational setup than in the original proof, in particular involving only (families of) one-dimensional arithmetic progressions, as opposed to multidimensional arithmetic progressions. Thirdly, we abstract the analysis of limiting densities along the (now one-dimensional) arithmetic progressions by introducing the notion of a family of arithmetic progressions with the ``double counting property''. We also present a simplified version of the argument that is capable of establishing Roth's theorem on arithmetic progressions of length three.

中文翻译：

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Szemerédi 对 Szemerédi 定理的证明

1975 年，Szemeredi 提出了著名的结论，即任何一组上密度为正的整数都包含任意长的等差数列。证明极其复杂但很基本，所需的主要工具是范德瓦尔登定理和现在称为 Szemeredi 正则性引理的引理，以及对沿多维集合的极限密度的精细分析（最终基于重复计算论证）算术级数。在本说明中，我们介绍了该证明的一种安排，其中包含了许多符号和技术简化。首先，我们用 Frieze 和 Kannan 的更简单的“弱正则性引理”代替正则性引理的使用。其次，我们提取了 Szemeredi 证明核心的关键归纳步骤（称为“引理 5”、“引理 6” '，以及那篇论文中的“事实 12”）作为独立的定理，可以用比原始证明中更少的符号设置来表述，特别是只涉及一维等差数列的（族），而不是多维算术级数。第三，我们通过引入具有“双重计数特性”的等差数列族的概念，抽象了沿（现在是一维的）等差数列的极限密度的分析。我们还提出了一个简化版本的论证，它能够建立关于长度为 3 的等差数列的 Roth 定理。特别是仅涉及（族）一维等差数列，而不是多维等差数列。第三，我们通过引入具有“双重计数特性”的等差数列族的概念，抽象了沿（现在是一维的）等差数列的极限密度的分析。我们还提出了一个简化版本的论证，它能够建立关于长度为 3 的等差数列的 Roth 定理。特别是仅涉及（族）一维等差数列，而不是多维等差数列。第三，我们通过引入具有“双重计数特性”的等差数列族的概念，抽象了沿（现在是一维的）等差数列的极限密度的分析。我们还提出了一个简化版本的论证，它能够建立关于长度为 3 的等差数列的 Roth 定理。