有限海森堡群中的尖锐精细方向Kakeya估计
Sharp refined-direction Kakeya estimates in finite Heisenberg groups
AI总结：
本文针对有限海森堡群，证明了尖锐富方向Kakeya估计，确定了对应ℓᵘ→ℓᵛ估计中q的尖锐指数，证明结合多项式方法与重数及仿射辛群作用的概率覆盖论证。
AI中文摘要：
设n≥2，q为奇素数幂。本文首要目标是证明：对任意E⊂ℍₙ(𝔽_q)及任意λ>0，如下尖锐富方向估计成立：|{ϑ∈Dₙ: M^rd_{ℍₙ}1_E(ϑ)≥λ}|≲ₙ q^{2n−1}|E|λ^{−2n}。次要目标是确定对任意1≤u,v≤∞，对应ℍᵘ→ℍᵛ估计中q的尖锐指数，精确地证明Aⁿᵣᵈ(u,v)=max{(2n−1)/v,1−1/u,2n/v−1/u,1+2n/v−(2n+1)/u}。证明结合了多项式方法与重数，以及基于仿射辛群作用的概率覆盖论证。
英文摘要：
Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $λ>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geqλ \right\} \right| \lesssim_n q^{2n-1}|E|λ^{-2n}. \] The second aim is to determine, for every $1\leq u,v\leq\infty$, the sharp exponent of $q$ in the corresponding $\ell^u\to\ell^v$ estimate. More precisely, we prove that \[ A_n^{\mathrm{rd}}(u,v) = \max\left\{ \frac{2n-1}{v},\ 1-\frac1u,\ \frac{2n}{v}-\frac1u,\ 1+\frac{2n}{v}-\frac{2n+1}{u} \right\}. \] The proof combines the polynomial method with multiplicities and a probabilistic covering argument based on the action of the affine symplectic group.
