Restricted Marstrand's projection theorem for general families of linear subspaces

Abstract

This paper investigates a refinement of Marstrand's projection theorem; more specifically, let ${\Pi }_t,t\in [0,1]$ be a family of $m$ dimensional subspaces of the Euclidean space ${\mathbb{R}}^n$ and let $P_t:{\mathbb{R}}^4\mapsto {\Pi }_t$ be the orthogonal projections onto ${\Pi }_t$. We hope to determine the conditions on ${\Pi }_t$ 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 ${\Pi }_t$ 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.

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