Iterating the Big‐Pieces operator and larger sets

Abstract

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