A Marstrand projection theorem for lines

• Main Author: Gan, Shengwen
• Format: Journal Article
• Language: English
• Published: 26-10-2023
• Subjects: Mathematics - Classical Analysis and ODEs; Mathematics - Combinatorics; Mathematics - Metric Geometry
• DOI: 10.48550/arxiv.2310.17454

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.