A Marstrand projection theorem for lines (2310.17454v1)

Abstract: Fix integers $1<k<n$. For $V\in G(k,n)$, let $P_V: \mathbb{R}^n\rightarrow V$ be the orthogonal projection. For $V\in G(k,n)$, define the map [ \pi_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 $\boldsymbol{A} \subset A(1,n)$ with $\text{dim}(\boldsymbol{A})=a$, we have [ \text{dim}(\pi_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.