Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
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DOI：
10.1007/bf02813304
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发表时间：
1977-12
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影响因子：
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通讯作者：
H. Furstenberg
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文献类型：
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作者：
H. Furstenberg
van der Waerden（[2]）的一个著名结果指出，如果整数被划分为有限多个子集，其中一个子集具有具有任意有限长度的等差数列的性质。erds和Tur~ in推测密度正渐近整数的任意子集具有任意长度的等差数列。Roth用解析方法在b[5]中证明了长度为3的等差数列也是如此。Szemer6di在对长度为4（[6]）的等差数列得到初步结果后，最终证明了erds在[7]中的猜想。szemerdi的方法是组合的，他利用了van der Waerden的定理。我们将给出这个结果的另一种证明，说明它是如何从我们将要表述的塞默尔迪定理的遍历理论版本推导出来的。然后我们分阶段证明了这个遍历理论定理；首先是在系统弱混合的情况下，然后是在一般情况下但是在与长度为3的等差数列的存在相对应的公式中。最后，在对一般遍历系统的结构作了一些初步的探讨之后，我们在一般情况下证明了该定理。
A well known result of van der Waerden ([2]) states that if the integers are partitioned into finitely many subsets, one of these has the property that it possesses arithmetic progressions of arbitrary finite length. ErdSs and Tur~ in conjectured that any subset of the integers of positive asymptotic density would possess arithmetic progressions of arbitrary length. Roth, using analytic methods showed in [5] that this was the case for arithmetic progressions of length 3. After a preliminary result for arithmetic progressions of length 4 ([6]), Szemer6di finally proved ErdSs' conjecture in [7]. Szemer6di's method is combinatorial and he makes use of van der Waerden's theorem. We shall present a different proof of this result by showing how it follows from an ergodic theoretic version of Szemer6di's theorem that we shall formulate. We then prove this ergodic theoretic theorem in stages; first in case the system is weakly mixing, and then in the general case but in the formulation that corresponds to the existence of arithmetic progressions of length three. Finally after some preliminaries regarding the structure of general ergodic systems we prove the theorem in the general case.