Restricted Marstrand's projection theorem for general families of linear subspaces
Jiahan Du
TL;DR
The paper addresses refining Marstrand’s projection theorem for a family of projections $P_t$ onto $2$-dimensional subspaces $\Pi_t$ of $\mathbb{R}^4$, seeking conditions on analytic families that ensure $\dim_H P_t(A)=\min(\dim_H A,2)$ for a.e. $t$. It advances a partial result (Theorem Main) under a concrete polynomial-parameterized setup, proving a strict lower bound $\dim_H P_t(A)\!>\!1+\epsilon$ for all $A$ with $\dim_H A=2$, and develops a novel framework using a curved-tube polynomial Wolff axiom, polynomial partitioning, and a discretized finishing argument. The core methodology recasts projection problems as curved Kakeya-type problems, leveraging polynomial partitioning to obtain multilinear Kakeya bounds for curved tubes, and then employing broad–narrow analysis to pass to linear estimates. The results shed light on the structure of restricted projection phenomena and demonstrate how semialgebraic and algebraic-geometric techniques can drive progress in geometric measure theory for projection questions.
Abstract
This paper investigates a refinement of Marstrand's projection theorem; more specifically, let $Π_t, t\in[0,1]$ be a family of $m$ dimensional subspaces of the Euclidean space $\mathbb{R}^n$ and let $P_t:\mathbb{R}^4\mapsto Π_t$ be the orthogonal projections onto $Π_t$. We hope to determine the conditions on $Π_t$ under which, for any Borel $A\subset\mathbb{R}^n$, $\dim_H P_t(A)=\min(m,\dim_H A)$ holds for almost every $t$. We propose a conjectured condition on $Π_t$ and provide partial progress towards its resolution. We first establish a version of the polynomial Wolff axiom, and then apply polynomial partitioning to derive a version of the $L^p$ Kakeya inequality. Finally, we use a discretization procedure to obtain the desired bound.