Sharp refined-direction Kakeya estimates in finite Heisenberg groups

Let $n\ge 2$ 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 $\lambda >0$, the following sharp rich-direction estimate holds $$\mid \{\vartheta \in D_n:M_{{\mathbb{H}}_n}^{\mathrm{r}\mathrm{d}}{\mathbf{1}}_E(\vartheta )\ge \lambda \}\mid {\lesssim }_n{q}^{2n-1}\mathrm{\mid }E\mathrm{\mid }{\lambda }^{-2n}\mathrm{.}$$ The second aim is to determine, for every $1\le u,v\le \mathrm{\infty }$, the sharp exponent of $q$ in the corresponding ${\mathrm{\ell }}^u\to {\mathrm{\ell }}^v$ estimate. More precisely, we prove that $$A_n^{\mathrm{r}\mathrm{d}}(u,v)=\max \{\frac{2n-1}{v},1-\frac{1}{u},\frac{2n}{v}-\frac{1}{u},1+\frac{2n}{v}-\frac{2n+1}{u}\}\mathrm{.}$$ The proof combines the polynomial method with multiplicities and a probabilistic covering argument based on the action of the affine symplectic group.