How Many Reflections Make a Dihedral Set Large?

TL;DR

This paper investigates the maximum size of product sets formed from subsets of the infinite dihedral group, revealing how the number of reflections affects growth and providing explicit formulas and asymptotic behavior.

Contribution

It provides explicit formulas and asymptotic analysis for the maximum size of product sets in the infinite dihedral group, highlighting the role of reflections.

Findings

Explicit formula for maximum size of S^n given subset size and reflections.

Optimal number of reflections for maximizing |S^n|.

Asymptotic expression for large n and fixed k.

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~$k=~n$.