AI 中文总结
针对指数为p的海森堡群H_{p^3}，通过阶值增长定理等工具证明其小达文波特常数𝖽(H_{p^3})=3p-3，给出了上界与下界的统一证明。
AI 中文摘要
设p为奇素数，H_{p^3}=UT₃(𝔽ₚ)是阶为p³且指数为p的海森堡群。我们证明𝖽(H_{p^3})=3p-3。证明的主要要素是一个阶值增长定理：若B是𝔽ₚ²中n个非零向量的非共线零和序列，则对B排序后得到的交替面积至少取到min(p,n-1)个不同值。该定理的证明采用短收缩归纳法：收缩一对合适的独立向量，在两种排序中替换被收缩的向量，并应用柯西-达文波特定理。多项式相对子和定理与严格表示刚性引理随后将这种局部增长转化为统一的扩散界；结合H_{p^3}的标准乘积-1准则，该扩散界给出上界，而常规序列x^{p-1}y^{p-1}v^{p-1}给出下界。
英文摘要
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.
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