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 $>1$. Let $Z \subset \mathbb{R}^{n+1}$ be analytic and let $0 < s < \min{\dim Z, 1}$. Then \begin{equation*} \dim {z \in \Sigma : \dim (Z \cdot z) < s} \le (n-2)+s = (n-1) + (s-1) < 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 K\"{a}enm\"{a}ki-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 H\'{e}ra, H\'{e}ra-Keleti-M\'{a}th\'{e} and D{\k{a}}browski-Orponen-Villa. Moreover, our method to show the restricted projection theorem can also give a new proof for the Mattila's projection theorem in $\mathbb{R}^n$ with $n \ge 3$.