Tangent points of lower content $d$-regular sets and $\beta$ numbers

TL;DR

This paper characterizes tangent points of lower content $d$-regular sets in Euclidean space using a Dini-type condition on Jones $eta$ numbers, even without assuming $\sigma$-finiteness of the measure.

Contribution

It establishes a new connection between tangent points and $eta$ numbers for sets with non-$\sigma$-finite Hausdorff measure, using a modified $eta$ coefficient based on Hausdorff content.

Findings

Identifies tangent points via a Dini condition on $eta$ numbers.

Extends previous results to sets without $\sigma$-finite measure.

Introduces a variant of $eta$ coefficients using Hausdorff content.

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.