Fourier integral operators, fractal sets and the regular value theorem

TL;DR

This paper establishes a dimension estimate for level sets of regular functions on large fractal sets in Euclidean space, using Fourier integral operators and connecting to the Falconer distance conjecture.

Contribution

It introduces a new approach to bounding the Minkowski dimension of certain level sets on fractal sets via Fourier integral operators, extending regular value theorems to fractal contexts.

Findings

Upper Minkowski dimension of level sets is bounded by the Hausdorff dimension minus the number of equations.

Results are sharp, with failure of the inequality below certain Hausdorff dimension thresholds.

Connections made with lattice point distribution on convex surfaces and combinatorial geometry.

Abstract

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{{\mathcal H}}({\mathcal E})$, and $\phi$ is a sufficiently regular function, then the upper Minkowski dimension of the set $$ \{w \in {\mathcal E}: \phi_l(w)=t_l; 1 \leq l \leq m \}$$ does not exceed $dim_{{\mathcal H}}({\mathcal E})-m$, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier Integral Operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are in general sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.