Restricted projections to lines in $\mathbb{R}^{n+1}$

Abstract We prove the following restricted projection theorem. Let $n\ge 3$ and $\Sigma \subset S^{n}$ be an $(n-1)$ -dimensional $C^2$ manifold such that $\Sigma$ has sectional curvature $\gt1$ . Let $Z \subset \mathbb{R}^{n+1}$ be analytic and let $0 \lt s \lt \min\{\dim Z, 1\}$ . Then \begin{equation*} \dim \{z \in \Sigma\; :\; \dim (Z \cdot z) \lt s\} \le (n-2)+s = (n-1) + (s-1) \lt n-1.\end{equation*} In particular, for almost every $z \in \Sigma$ , $\dim (Z \cdot z) = \min\{\dim Z, 1\}$ . The core idea, originated from Käenmäki–Orponen–Venieri, is to transfer the restricted projection problem to the study of the dimension lower bound of Furstenberg sets of cinematic family contained in $C^2([0,1]^{n-1})$ . This cinematic family of functions with multivariables are extensions of those of one variable by Pramanik–Yang–Zahl and Sogge. Since the Furstenberg sets of cinematic family contain the affine Furstenberg sets as a special case, the dimension lower bound of Furstenberg sets improves the one by Héra, Héra–Keleti–Máthé and Dąbrowski–Orponen–Villa. Moreover, our method to show the restricted projection theorem can also give a new proof for the Mattila projection theorem in $\mathbb{R}^n$ with $n \ge 3$ .