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Partial associativity and rough approximate groups
Geometric and Functional Analysis ( IF 2.4 ) Pub Date : 2020-11-08 , DOI: 10.1007/s00039-020-00553-1
W. T. Gowers , J. Long

Suppose that a binary operation \(\circ \) on a finite set X is injective in each variable separately and also associative. It is easy to prove that \((X,\circ )\) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples \((x,y,z)\in X^3\) satisfy the equation \(x\circ (y\circ z)=(x\circ y)\circ z\). Other results in additive combinatorics would lead one to expect that there must be an underlying ‘group-like’ structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.



中文翻译:

部分关联性和粗糙近似群

假设对有限集X的二元运算\(\ circ \)在每个变量中分别是内射的,并且也是关联的。很容易证明\((X,\ circ)\)必须是一个组。在本文中,我们研究了如果仅知道三元组\((x,y,z)\在X ^ 3 \中)满足方程\(x \ circ(y \ circ z)=( x \ circ y)\ circ z \)。加性组合法的其他结果会使人们期望,必须有一个潜在的“类群”结构负责大量的关联三元组。我们证明确实是这样:乘法表中必须有一个比例大小的子集,该子集与度量标准组的乘法表的一部分大致相符。Green的最新结果表明,这种度量近似是必要的:并非总能获得与组的乘法表的一部分相符的比例大小的子集。

更新日期:2020-11-09
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