The infinite dihedral group is generated by two reflections. Give it a finite subset S of size k and ask: how large can the n-fold product set S^n become? The answer depends not just on k and n but on the composition of S — specifically, how many of the k elements are reflections versus translations.
Greenfeld et al. (arXiv:2603.22533) derive an explicit formula for the maximum size of S^n as a function of the reflection count. The formula is exact, not asymptotic. As n grows with k fixed, the number of reflections affects the growth rate through a multiplicative coefficient that admits a clean probabilistic interpretation — it equals the probability that a certain random walk returns to its starting configuration.
The result separates two contributions to product set growth that are usually entangled: the type of generator (reflection vs. translation) and the rate of expansion. In most groups, these interact in complicated ways because different generator types combine to produce new elements at different rates depending on the group's geometry. In the infinite dihedral group, the interaction factors cleanly — the reflection count determines a coefficient, and the translation count determines the base of exponential growth.
This factorization matters because it provides a template for understanding product set growth in groups where reflections and translations are fundamental (Coxeter groups, wallpaper groups, crystallographic groups). The clean separation suggests that the apparently complex interaction between generator types can sometimes be resolved into independent multiplicative contributions.
The through-claim: in the infinite dihedral group, generator composition and growth rate decouple. The number of reflections scales the product set by a probabilistic factor without altering its exponential character — a surprisingly clean structure in a problem that could have been combinatorially messy.