Products of normal subsets

In this paper we consider which families of finite simple groups G G have the property that for each ϵ > 0 \epsilon > 0 there exists N > 0 N > 0 such that, if | G | ≥ N |G| \ge N and S , T S, T are normal subsets of G G with at least ϵ | G | \epsilon |G| elements each, then every non-trivial element of G G is the product of an element of S S and an element of T T . We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form P S L n ( q ) \mathrm {PSL}_n(q) where q q is fixed and n → ∞ n\to \infty . However, in the case S = T S=T and G G alternating this holds with an explicit bound on N N in terms of ϵ \epsilon . Related problems and applications are also discussed. In particular we show that, if w 1 , w 2 w_1, w_2 are non-trivial words, G G is a finite simple group of Lie type of bounded rank, and for g ∈ G g \in G , P w 1 ( G ) , w 2 ( G ) ( g ) P_{w_1(G),w_2(G)}(g) denotes the probability that g 1 g 2 = g g_1g_2 = g where g i ∈ w i ( G ) g_i \in w_i(G) are chosen uniformly and independently, then, as | G | → ∞ |G| \to \infty , the distribution P w 1 ( G ) , w 2 ( G ) P_{w_1(G),w_2(G)} tends to the uniform distribution on G G with respect to the L ∞ L^{\infty } norm.