A Marstrand projection theorem for lines

Abstract

Fix integers $1<k<n$. For $V\in G(k,n)$, let $P_V:{\mathbb{R}}^n\to V$ be the orthogonal projection. For $V\in G(k,n)$, define the map \[ π_V: A(1,n)\rightarrow A(1,V)\bigsqcup V. \] \[ \ell\mapsto P_V(\ell). \] For any $0<a<\text{dim}(A(1,n))$, we find the optimal number $s(a)$ such that the following is true. For any Borel set $\mathbf{A}\subset A(1,n)$ with $\text{dim}(\mathbf{A})=a$, we have \[ \text{dim}(π_V(\boldsymbol{A}))=s(a), \text{for a.e. } V\in G(k,n). \] When $A(1,n)$ is replaced by $A(0,n)={\mathbb{R}}^n$, it is the classical Marstrand projection theorem, for which $s(a)=\min \{k,a\}$. A new ingredient of the paper is the Fourier transform on affine Grassmannian.

25 pages; 2 figures