Restricted projections in positive characteristic via Fourier extension and restriction estimates

Let $d\ge3$ and $\mathbb{F}_q^{\,d}$ be the $d$-dimensional vector space over a finite field of order $q$, where $q$ is an odd prime power. Let $X_\pi$ be the set of lines through the origin intersecting the slice $\pi\cap S^{d-1}$, where $\pi=\{x_d=\lambda\}$ and $S^{d-1}=\{x:\|x\|=1\}$. For $E\subset\mathbb{F}_q^{\,d}$ and $N\ge1$, we study the exceptional sets $$T_1(X_\pi,E,N)=\bigl\{V\in X_\pi:\ |\pi_V(E)|\le N\bigr\},\qquad T_2(X_\pi,E,N)=\bigl\{V\in X_\pi:\ |\pi_{V^\perp}(E)|\le N\bigr\},$$ with their respective natural ranges of $N$. Using discrete Fourier analysis together with restriction/extension estimates for cone and sphere-type quadrics over finite fields, we obtain sharp upper bounds (up to constant factors) for $\lvert T_1\rvert$ and $\lvert T_2\rvert$, with separate analyses for the cases $\lambda \in \{0, \pm 1\}$. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of $\lambda^{2}-1$ and the parity of $d$ determine the size of the exceptional sets. As an application, when $|E|\ge q$, there exists a positive proportion of elements $\mathbf{y}\in \pi\cap S^{d-1}$ such that the pinned dot-product sets $\{\mathbf{y}\cdot \mathbf{x}\colon \mathbf{x}\in E\}$ have cardinality $\Omega(q)$. We further study analogous families arising from the spheres of radii $0$ and $-1$, and, by combining the results, recover the known estimates for projections over the full Grassmannian, complementing a result of Chen (2018).