Strong exceptional parameters for the dimension of nonlinear slices

Abstract

Let $1\le m<s\le n$ 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 $({\hat{I}}_{\hat{I}}»:\hat{I}\text{c◯}\to {\mathbb{R}}^m{)}_{\hat{I}»\in U}$ is a family of generalized projections that satisfies the transversality and strong regularity conditions of degree $0$, then for every $A\subseteq \hat{I}\text{c◯}$ with ${\mathcal{H}}^s(A)>0$, the set of $\hat{I}»$ in the parameter space $U\subseteq {\mathbb{R}}^N$ such that $\dim (A\cap {\hat{I}}_{\hat{I}}{»}^{-1}(x))<s-m$ 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 $({\hat{I}}_{\hat{I}}»{)}_{\hat{I}»\in U}$ is only transversal and strongly regular of some sufficiently small order ${\hat{I}}^2>0$, a similar conclusion holds modulo an error term of order $β^{1/3}$.