A Uniform Proof for the Small Davenport Constant of the Exponent-$p$ Heisenberg Group
Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^2$, then the alternating areas obtained by ordering $B$ assume at least $\min(p,n-1)$ distinct values. Its proof is a short contraction induction: contract a suitable independent pair, replace the contracted vector in both orders, and apply Cauchy--Davenport. A polynomial relative-subsum theorem and a sharp representation-rigidity lemma then turn this local growth into a uniform spread bound. Combined with the standard product-one criterion for $H_{p^3}$, the spread bound yields the upper bound; the usual sequence $x^{p-1}y^{p-1}v^{p-1}$ gives the lower bound.