Strong exceptional parameters for the dimension of nonlinear slices
Ryan E. G. Bushling
TL;DR
The paper establishes a strong, quantitative Marstrand-type slicing theorem for nonlinear slices generated by a transversely interacting family of generalized projections, within the Peres–Schlag framework. By embedding the slice problem into a Fourier-analytic setting via the surrogate map $\Psi_{\lambda}$ and proving a central main lemma, it yields explicit bounds on the Hausdorff dimension of the exceptional parameter set and, when $\mathcal{H}^s(A)<\infty$, a universality across all positive-measure subsets of $A$. The methodology hinges on energy estimates, a power-series divergence argument, and a Falconer–Mattila universalization step, and it extends to applications in the vertical geometry of the Heisenberg group through isotropic projections. This provides a robust, high-precision framework for understanding how typical fibers of nonlinear projections intersect sets of fractional dimension, with concrete implications for non-Euclidean slicing problems. The results sharpen our understanding of when slices retain the expected dimension and quantify the exceptional-parameter phenomena with explicit $\beta$-dependent corrections.
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 $(Π_λ: Ω\to \mathbb{R}^m)_{λ\in U}$ is a family of generalized projections that satisfies the transversality and strong regularity conditions of degree $0$, then for every $A \subseteq Ω$ with $\mathcal{H}^s(A) > 0$, the set of $λ$ in the parameter space $U \subseteq \mathbb{R}^N$ such that $\dim\!\big(A \cap Π_λ^{-1}(x)\big) < s-m$ 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 $(Π_λ)_{λ\in U}$ is only transversal and strongly regular of some sufficiently small order $β> 0$, a similar conclusion holds modulo an error term of order $β^{1/3}$.