Strong exceptional parameters for the dimension of nonlinear slices

Abstract

Let $1 \leq m < s \leq n$ and let $A \subseteq \mathbb{R}^n$ be a Borel set of with $s$-dimensional Hausdorff measure $\mathcal{H}^s(A) > 0$. The classical Marstrand slicing theorem states that, for almost every $m$-dimensional subspace $V \subset \mathbb{R}^n$, there is a positive-measure set of $x \in V$ such that $x + V^\perp$ intersects $A$ in a set of Hausdorff dimension $s-m$. We prove a strong and quantitative version of Marstrand's slicing theorem in the Peres-Schlag framework. In particular, if $(\Pi_\lambda: \Omega \to \mathbb{R}^m)_{\lambda \in U}$ is a family of generalized projections that satisfies the transversality and strong regularity conditions of degree $0$, then for every $A \subseteq \Omega$ with $\mathcal{H}^s(A) > 0$, the set of $\lambda$ in the parameter space $U \subseteq \mathbb{R}^N$ such that $\dim\!\big(A \cap \Pi_\lambda^{-1}(x)\big) < s-m$ for a.e. $x \in \mathbb{R}^m$ has Hausdorff dimension at most $N + m - s$. If moreover $\mathcal{H}^s(A) < \infty$, then this exceptional set is universal for the subsets of $A$ with positive $s$-dimensional Hausdorff measure in the sense that this same collection of parameters contains the corresponding exceptional sets of all those subsets of $A$. When $(\Pi_\lambda)_{\lambda \in U}$ is only transversal and strongly regular of some sufficiently small order $\beta > 0$, a similar conclusion holds modulo an error term of order $\beta^{1/3}$.