Commuting Probability for Approximate Subgroups of a Finite Group

For subsets $X,Y$ of a finite group G, we write Pr (X,Y) for the probability that two random elements x∈ X and y∈ Y commute. This paper addresses the relation between the structure of an approximate subgroup A⊆ G and the probabilities Pr (A,G) and Pr (A,A). The following results are obtained. Theorem 1.1: Let A be a K-approximate subgroup of a finite group G, and let Pr (A,G)≥ε>0. There are two (ε, K)-bounded positive numbers γ and K0 such that G contains a normal subgroup T and a K0-approximate subgroup B such that |A∩ B|≥ γ\; max\|A|,|B|\ while the index $[G:T]$ and the order of the commutator subgroup [T, B ] are (ε, K)-bounded. Theorem 1.2: Let A be a K-approximate subgroup of a finite group G, and let Pr (A,A)≥ε>0. There are two (ε, K)-bounded positive numbers γ and s and a subgroup C≤ G such that |C ∩ A²| > γ |A| and |C^|≤ s. In particular, A is contained in the union of at most γ⁻¹K² left cosets of the subgroup C. It is also shown that the above results admit approximate converses.