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.
Categories: math.CA