Quantitative Equidistribution for Certain Quadruples in Quasi-Random Groups

Bergelson and Tao have recently proved that if G is a D -quasi-random group, and x, g are drawn uniformly and independently from G , then the quadruple ( g, x, gx, xg ) is roughly equidistributed in the subset of G 4 defined by the constraint that the last two coordinates lie in the same conjugacy class. Their proof gives only a qualitative version of this result. The present note gives a rather more elementary proof which improves this to an explicit polynomial bound in D −1 .