Transversal families of nonlinear projections and generalizations of Favard length

TL;DR

This paper demonstrates that various nonlinear projection operators satisfy transversality conditions, enabling new quantitative estimates for decay rates of Favard-type lengths and related measures, with applications to fractal sets.

Contribution

It establishes that several classes of nonlinear projections are transversal, leading to new bounds on Favard length variants and their decay rates.

Findings

Nonlinear projections satisfy transversality conditions.

Quantitative lower bounds for decay rates of Favard length variants.

Simplified proof for Favard curve length decay of Cantor sets.

Abstract

Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a \textit{transversality condition}. In this work, we establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor.