Journal
GEOMETRIC AND FUNCTIONAL ANALYSIS
Volume 30, Issue 6, Pages 1583-1647Publisher
SPRINGER BASEL AG
DOI: 10.1007/s00039-020-00553-1
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Suppose that a binary operation. on a finite set X is injective in each variable separately and also associative. It is easy to prove that (X, omicron) 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) is an element of X-3 satisfy the equation x omicron (y omicron z) = (x omicron y) omicron 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.
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