Iterating the Big--Pieces operator and larger sets

TL;DR

This paper demonstrates that iterating the Big Pieces operator on Ahlfors-David regular sets preserves their structure, showing stability after two iterations and extending results to metric spaces with a broader family of sets.

Contribution

It proves that BP(BP(LG)) sets are contained in BP(LG) sets in metric spaces, generalizing Euclidean results and simplifying proofs of stability after two iterations.

Findings

BP(BP(LG)) sets are contained in BP(LG) sets.

Stability of the BP operator after two iterations.

Results extend to metric spaces with Ahlfors-David regular sets.

Abstract

We show that if an Ahlfors-David regular set $E$ of dimension $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by $BP(BP(LG))$), then $E\subset \tilde{E}$ where $\tilde{E}$ is Ahlfors-David regular of dimension $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by $BP(LG)$. Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors-David regular sets $\mathcal{F}$ replacing $LG$. A simple corollary is the stability of the BP operator after 2 iterations. This was previously only known in the Euclidean setting for the case $\mathcal{F}= LG$ with substantially more complicated proofs.