2020/06/30 by Matthew Hyde, Hyde, Matthew
Mathematics · #28A12 #28A75 #28A78 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Point processes and geometric inequalities #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2006.16677
openalex publication_date 2020/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In his 1990 paper, Jones proved the following: given E ⊆ ℝ2, there exists a curve Γ such that E ⊆ Γ and \mathscrH1(Γ) ∼ diam E + ∑Q βE(3Q)2ℓ(Q). Here, βE(Q) measures how far E deviates from a straight line inside Q. This was extended by Okikiolu to subsets of ℝn and by Schul to subsets of a Hilbert space. In 2018, Azzam and Schul introduced a variant of the Jones β-number. With this, they, and separately Villa, proved similar results for lower regular subsets of ℝn. In particular, Villa proved that, given E ⊆ ℝn which is lower content regular, there exists a `nice' d-dimensional surface F such that E ⊆ F and \mathscrHd(F) ∼ diam( E)d + ∑Q βE(3Q)2ℓ(Q)d. In this context, a set F is `nice' if it satisfies a certain topological non degeneracy condition, first introduced in a 2004 paper of David. In this paper we drop the lower regularity condition and prove an analogous result for general d-dimensional subsets of ℝn. To do this, we introduce a new d-dimensional variant of the Jones β-number that is defined for any set in ℝn.