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 Σ: \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 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 D{ą}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 $n\ge 3$.

37 pages, 2 figures