How Many Reflections Make a Dihedral Set Large?

Abstract

Given a size-$k$ subset $S$ of a group $G$, how large can the product set $S^n$ be? We study this question, at several layers of refinement, for the infinite dihedral group. First, we give an explicit formula for the maximum size of $S^n$ among all size-$k$ subsets with a prescribed number of reflections. We then determine the optimal number of reflections that a size-$k$ set should contain in order to maximize $|S^n|$. When $k$ is fixed and $n\to\infty$, we obtain a clean asymptotic expression for the maximal size of $S^n$. Moreover, we compute this asymptotic separately for each fixed number of reflections in $S$. We show that the number of reflections influences the asymptotic size of $S^n$ only through a multiplicative coefficient, which admits a direct probabilistic interpretation. Finally, we compute the growth exponent of the maximum of $|S^n|$ when~$k=~n$.