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Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

- Matthew Badger
- Mathematics
- 23 March 2010

We show the David–Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of… Expand

Multiscale analysis of 1-rectifiable measures: necessary conditions

- Matthew Badger, R. Schul
- Mathematics
- 2 July 2013

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $$\mathbb {R}^n$$Rn, $$n\ge 2$$n≥2. To each locally finite Borel measure $$\mu… Expand

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

- Matthew Badger, R. Schul
- Mathematics
- 11 February 2016

Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean… Expand

Two sufficient conditions for rectifiable measures

- Matthew Badger, R. Schul
- Mathematics
- 29 December 2014

We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extending… Expand

Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

- Matthew Badger
- Mathematics, Computer Science
- J. Lond. Math. Soc.
- 7 September 2011

We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away from a hyperplane in… Expand

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

- M. Akman, Matthew Badger, S. Hofmann, J. M. Martell
- Mathematics
- 8 July 2015

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that… Expand

Structure of sets which are well approximated by zero sets of harmonic polynomials

- Matthew Badger, Max Engelstein, T. Toro
- Mathematics
- 10 September 2015

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries… Expand

Geometry of Measures in Real Dimensions via Hölder Parameterizations

- Matthew Badger, Vyron Vellis
- Mathematics
- 23 June 2017

We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $$\mathbb {R}^n$$Rn when s is a real number between 0 and n. This topic… Expand

LOCAL SET APPROXIMATION: MATTILA–VUORINEN TYPE SETS, REIFENBERG TYPE SETS, AND TANGENT SETS

- Matthew Badger, Stephen Lewis
- Mathematics
- Forum of Mathematics, Sigma
- 27 September 2014

We investigate the interplay between the local and asymptotic geometry of a set $A\subseteq \mathbb{R}^{n}$ and the geometry of model sets ${\mathcal{S}}\subset {\mathcal{P}}(\mathbb{R}^{n})$, which… Expand

Generalized rectifiability of measures and the identification problem

- Matthew Badger
- Mathematics
- Complex Analysis and its Synergies
- 27 March 2018

One goal of geometric measure theory is to understand how measures in the plane or a higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises… Expand

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