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Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited
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 ofExpand
Multiscale analysis of 1-rectifiable measures: necessary conditions
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 $$\muExpand
Multiscale Analysis of 1-rectifiable Measures II: Characterizations
Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional EuclideanExpand
Two sufficient conditions for rectifiable measures
We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extendingExpand
Flat points in zero sets of harmonic polynomials and harmonic measure from two sides
  • Matthew Badger
  • Mathematics, Computer Science
  • J. Lond. Math. Soc.
  • 7 September 2011
We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away from a hyperplane inExpand
Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume thatExpand
Structure of sets which are well approximated by zero sets of harmonic polynomials
The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundariesExpand
Geometry of Measures in Real Dimensions via Hölder Parameterizations
We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $$\mathbb {R}^n$$Rn when s is a real number between 0 and n. This topicExpand
We investigate the interplay between the local and asymptotic geometry of a set $A\subseteq \mathbb{R}^{n}$ and the geometry of model sets ${\mathcal{S}}\subset {\mathcal{P}}(\mathbb{R}^{n})$, whichExpand
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 arisesExpand