Products of normal subsets

Abstract:

In this paper we consider which families of finite simple groups $G$ have the property that for each $\epsilon > 0$ there exists $N > 0$ such that, if $|G| \ge N$ and $S, T$ are normal subsets of $G$ with at least $\epsilon |G|$ elements each, then every non-trivial element of $G$ is the product of an element of $S$ and an element of $T$.

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 $\mathrm {PSL}_n(q)$ where $q$ is fixed and $n\to \infty$. However, in the case $S=T$ and $G$ alternating this holds with an explicit bound on $N$ in terms of $\epsilon$.

Related problems and applications are also discussed. In particular we show that, if $w_1, w_2$ are non-trivial words, $G$ is a finite simple group of Lie type of bounded rank, and for $g \in G$, $P_{w_1(G),w_2(G)}(g)$ denotes the probability that $g_1g_2 = g$ where $g_i \in w_i(G)$ are chosen uniformly and independently, then, as $|G| \to \infty$, the distribution $P_{w_1(G),w_2(G)}$ tends to the uniform distribution on $G$ with respect to the $L^{\infty }$ norm.