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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

## Harmonic polynomials and tangent measures of harmonic measure

- Matthew Badger
- Mathematics
- 14 October 2009

We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric information for… 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
- MathematicsJournal of the London Mathematical Society
- 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

## 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

## 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

## Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

- Matthew Badger, Lisa M. Naples, Vyron Vellis
- MathematicsAdvances in Mathematics
- 4 June 2018

## 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
- MathematicsForum 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

## BEURLING'S CRITERION AND EXTREMAL METRICS FOR FUGLEDE MODULUS

- Matthew Badger
- Mathematics
- 23 July 2012

For each 1 ≤ p < ∞, we formulate a necessary and sufficient condition for an admissible metric to be extremal for the Fuglede p-modulus of a system of measures. When p = 2, this characterization… Expand

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