Greedy approximation algorithms for sparse collections

TL;DR

This paper introduces a greedy algorithm to approximate the Carleson constant of set collections, providing structural insights and applications to axis-parallel rectangles, with implications for sparse collection characterization.

Contribution

It presents a new greedy approximation method for the Carleson constant, along with structural theorems and applications to geometric set collections, improving understanding of sparse collections.

Findings

The algorithm approximates the Carleson constant with logarithmic loss.

Every collection can be partitioned into a linear number of sparse subfamilies based on its Carleson constant.

Characterization of the Carleson constant using only $L^{1, abla}$ estimates.

Abstract

We describe a greedy algorithm that approximates the Carleson constant of a collection of general sets. The approximation has a logarithmic loss in a general setting, but is optimal up to a constant with only mild geometric assumptions. The constructive nature of the algorithm gives additional information about the almost-disjoint structure of sparse collections. As applications, we give three results for collections of axis-parallel rectangles in every dimension. The first is a constructive proof of the equivalence between Carleson and sparse collections, first shown by H\"anninen. The second is a structure theorem proving that every collection $\mathcal{E}$ can be partitioned into $\mathcal{O}(N)$ sparse subfamilies where $N$ is the Carleson constant of $\mathcal{E}$. We also give examples showing that such a decomposition is impossible when the geometric assumptions are dropped. The third application is a characterization of the Carleson constant involving only $L^{1,\infty}$ estimates.