Associativity testing

Suppose we have a cancellative binary associative operation * on a finite set X. We say that it is delta-associative if the proportion of triples x, y, z such that x*(y*z) = (x*y)*z is at least delta.

Gowers and Long studied somewhat associative operations, and we will describe their main result. They also suggested an example of a somewhat associative operation which they conjectured does not (in a sense I will make precise) resemble a group operation. I will describe a proof of their conjecture, which uses a surprisingly large number of tools from additive combinatorics and group theory.