Characterising the big pieces of Lipschitz graphs property using projections

TL;DR

This paper characterizes when large subsets of Ahlfors-David regular sets in Euclidean space can be contained in Lipschitz graphs based on their projection properties, addressing a question posed by David and Semmes.

Contribution

It establishes a new criterion linking projection properties in L^2 to the big pieces of Lipschitz graphs property for Ahlfors-David regular sets.

Findings

Large subsets with many L^2 projections are contained in Lipschitz graphs

Provides a partial answer to a question by David and Semmes

Connects projection properties to geometric structure of sets

Abstract

We characterise the big pieces of Lipschitz graphs property in terms of projections. Roughly speaking, we prove that if a large subset of an $n$-Ahlfors-David regular set $E \subset \mathbb{R}^d$ has plenty of projections in $L^{2}$, then a large part of $E$ is contained in a single Lipschitz graph. This is closely related to a question of G. David and S. Semmes.