Plenty of big projections imply big pieces of Lipschitz graphs

Abstract

I prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

A note on the Hausdorff dimension of uniformly non flat sets with plenty of big projections

“…Since E R is Ahlfors d-regular (Lemma 3.2) and has PBP then, by the main result of [Orp21], it has BPLG. In particular, it is uniformly rectifiable.…”

“…In this note we record some consequences of a recent result by T. Orponen [Orp21], which states that an Ahlfors regular set with plenty of big projections (PBP) has big pieces of Lipschitz graph (BPLG). Let us give some definitions.…”

Rectifiability; a survey

“…David and Semmes call this property as E having big pieces of Lipschitz images of R m . There is a similar characterization by bi-Lipschitz images but big pieces of Lipschitz graphs (graphs over m-planes) is strictly stronger by an unpublished Venetian blind construction of Hrycak, see [Azz21] or [Orp21]. However, iterating this, big pieces of big pieces of Lipschitz graphs is equivalent to uniform rectifiability, see [AS12].…”

“…However, this does not seem to relate to David-Semmes uniform rectifiability. But we have the following theorem due to Orponen [Orp21]: Theorem 6.13. If E ⊂ R n is closed and AD-m-regular, then E has big pieces of Lipschitz graphs if and only there is θ > 0 such that for every x ∈ E and 0 < r < d(E) there is V ∈ G(n, m) for which…”

Structure of sets with nearly maximal Favard length

“…However, it remains plausible that the assumption FavpEq ě δ is sufficient to guarantee a quantitative version of Besicovitch's theorem under the additional assumption that E is 1-Ahlfors regular, or satisfies other "multi-scale 1-dimensionality" hypotheses. For recent partial results, and more discussion on this question, see [8,17,21,24]. The problem is closely related to Vitushkin's conjecture [25] on the connection between analytic capacity and Favard length, see [6,9].…”