Exceptional sets for restricted families of projections in $\mathbb{F}_q^d$

Abstract: Let $d\ge3$ and $\mathbb{F}<em>q<sup>{d}$ be the $d$-dimensional vector space over a finite field of order $q$, where $q$ is a prime power. Fix a slice $\pi={x_d=\lambda}$ of the unit sphere $S^{{d-1}={x\colon} ||x||=1}$ and let $X</em>\pi$ be the set of lines through the origin meeting $\pi\cap S<sup>{d-1}$. For $E\subset\mathbb{F}<em>q<sup>{d}$ and $N\ge1$, we study the exceptional sets [ \mathcal{T}_1(X\pi,E,N)=\bigl{V\in X_\pi:\ |\pi_V(E)|\le N\bigr},\qquad \mathcal{T}2(X\pi,E,N)=\bigl{V\in X_\pi:\ |\pi_{V^\perp}(E)|\le N\bigr}, ] on 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 bounds (up to constant factors) for $\lvert \mathcal{T}<em>1\rvert$ and $\lvert \mathcal{T}_2\rvert$, with separate treatment of the special slices $\lambda=\pm1$ and of the isotropic slice $\lambda=0$. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of $\lambda<sup>{2}-1$ 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 $\mathbf{y}\in X</em>\pi$ such that the pinned dot-product sets ${\mathbf{y}\cdot \mathbf{x}\colon \mathbf{x}\in E}$ are of cardinality $\Omega(q)$. We further treat analogous families arising from the spheres of radii $0$ and $-1$, and by combining these slices, recover the known estimates for projections over the full Grassmannian, complementing a result of Chen (2018).