[Paper Review] Tangent points of lower content $d$-regular sets and $\beta$ numbers

Michele Villa|arXiv (Cornell University)|Dec 7, 2017

Mathematical Dynamics and Fractals26 references3 citations

TL;DR

This paper establishes the equivalence between the finiteness of a β-number integral and the existence of tangent planes for lower content d-regular sets in R^n, proving that Hd-almost every point in such a set is a tangent point if and only if the L^2 integral of a content-based β coefficient is finite. The key contribution is extending Bishop and Jones' 1D result to higher dimensions under optimal integrability conditions on the β coefficient, using a Hausdorff content-based variant of Jones' β numbers to handle non-σ-finite Hausdorff measure.

ABSTRACT

Given a lower content $d$-regular set in $\mathbb{R}^n$, we prove that the subset of points in $E$ where a certain Dini-type condition on the so-called Jones $\beta$ numbers holds coincides with the set of tangent points of $E$, up to a set of $\mathcal{H}^d$-measure zero. The main point of our result is that $\mathcal{H}^d|_E$ is not assumed to be $\sigma$-finite; because of this, we use a certain variant of the $\beta$ coefficient, firstly introduced by Azzam and Schul in [AS1], which is given in terms of integration with respect to the Hausdorff content.

Motivation & Objective

• To characterize tangent points of lower content d-regular sets in R^n using a new variant of Jones' β numbers based on Hausdorff content.
• To resolve the challenge that Hd|E is not σ-finite, which invalidates standard rectifiability techniques.
• To extend the 1D result of Bishop and Jones (2011) on Jordan curves to d-regular sets in R^n for d ≥ 1.
• To establish the equivalence of four different definitions of tangent points (T1–T4) under optimal integrability conditions.
• To prove that the set of tangent points coincides with the set where the βp,d integral is finite, up to Hd-null sets.

Proposed method

• Introduces a content-based β coefficient βp,d_E(x,r) defined via integration against Hausdorff content, replacing the standard L∞ norm in Jones' original definition.
• Uses a Whitney-type decomposition of the ambient space into dyadic cubes and associated intrinsic cubes to localize analysis at small scales.
• Applies a variant of the Analyst’s Traveling Salesman Theorem for lower content d-regular sets, relying on the Azzam-Schul β coefficient framework.
• Employs a covering argument using coherent families of balls and planes, with careful control of distortion via Lipschitz constants and projections.
• Establishes a key estimate for the error term in the β coefficient approximation by bounding the distance from points in E to a candidate tangent plane using geometric control over Whitney cubes.
• Uses the fact that Hd(E) < ∞ and the lower content regularity condition to derive summability of β coefficients over dyadic cubes, leading to L^2 integrability.

Experimental results

Research questions

• RQ1 Under what conditions does the finiteness of the integral ∫₀^diam(E) βp,d_E(x,t)² dt/t imply the existence of a tangent plane at x ∈ E?
• RQ2 How can the concept of tangent points be consistently defined for sets where Hd|E is not σ-finite?
• RQ3 What is the optimal range of p for which the βp,d integral characterizes tangent points in d-regular sets for d ≥ 3?
• RQ4 How do the different definitions of tangent points (T1–T4) relate in the absence of σ-finiteness?
• RQ5 Can the 1D result of Bishop and Jones on Jordan curves be generalized to higher-dimensional d-regular sets?

Key findings

• For a bounded d-lower content regular set E ⊂ R^n, the set of points where ∫₀^diam(E) βp,d_E(x,t)² dt/t < ∞ has full Hd-measure if and only if the set of tangent points (in the sense of T2) has full Hd-measure.
• The equivalence between T1 (β integral finiteness), T2 (uniform distance to a plane), T3 (Hausdorff distance to a plane), and T4 (approximate tangent condition) holds up to Hd-null sets.
• For d = 1 or d = 2, the result holds for all 1 ≤ p < ∞; for d ≥ 3, it holds for 1 ≤ p ≤ 2d/(d−2), which is the optimal range for the L^p theory.
• The set E₀ ⊂ E of points satisfying the β integral condition is d-rectifiable, meaning it can be covered by countably many Lipschitz images of R^d up to Hd-null sets.
• The proof establishes that the error in approximating E by a tangent plane is controlled by the β coefficient, with the sum of β² over dyadic cubes being finite for Hd-a.e. x ∈ E.
• The result generalizes Bishop and Jones' 1999 theorem on Jordan curves to higher-dimensional d-regular sets, with the same integral condition characterizing tangent points.