Partial associativity and rough approximate groups

Abstract

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 $$(X,\circ )$$ ( X , ∘ ) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples $$(x,y,z)\in X^3$$ ( x , y , z ) ∈ X 3 satisfy the equation $$x\circ (y\circ z)=(x\circ y)\circ z$$ x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ 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.

Substructures in Latin squares

“…This is sometimes called the quadrangle condition (due to Brandt [6]). As explored by Gowers and Long [18], the number of cuboctahedra in a Latin square is a measure of "how associative" its corresponding loop is.…”

“…6 The reason for the name is that (as we will see in Figure 7.1), one can interpret these objects in such a way that they resemble geometric cuboctahedra. In [18], the authors simply call these objects "octahedra". 7 A quasigroup is a generalisation of a loop where we moreover do not demand the existence of an identity element.…”

On a conjecture of Gowers and Long

“…In a very interesting recent preprint [7], Gowers and Long considered somewhat associative binary operations, that is to say maps on a finite set which satisfy the associativity relation for a positive fraction of . Their main result is that such operations are all near (in a sense they make precise) to the group operation on a metric group .…”

“…It is natural to ask whether more might be true, namely whether such a binary operation must actually agree with a group operation a positive fraction of the time. In their paper, Gowers and Long present an example of an operation for which they conjecture (see [7, Conjecture 1.6]) that this is not the case. In view of their main result, the example is rather natural: one takes a natural and tractable example of a non‐abelian metric group , namely , and a large but finite subset .…”

“…As remarked above, Gowers and Long note that it seems very unlikely that any substantial portion of the multiplication table of can be embedded in a group operation, and make a precise conjecture, [7, Conjecture 1.6], to this effect. The following result establishes their conjecture.…”