A doubling measure on R^d can charge a rectifiable curve

  title={A doubling measure on R^d can charge a rectifiable curve},
  author={John B. Garnett and Rowan Killip and Raanan Schul},
For d > 2, we construct a doubling measure v on ℝ d and a rectifiable curve Γ such that ν(Γ) > 0. 

Figures from this paper

A function whose graph has positive doubling measure
We show that a doubling measure on the plane can give positive measure to the graph of a continuous function. This answers a question by Wang, Wen and Wen. Moreover we show that the doubling constant
On thin carpets for doubling measures
We study thin sets for doubling or isotropic doubling measures in R d \mathbb {R}^{d} . In our results we prove that the self-affine sets satisfying the open set condition with
Normal numbers are not fat for doubling measures
Sufficient conditions for C^1,α parametrization and rectifiability
  • Silvia Ghinassi
  • Mathematics
    Annales Academiae Scientiarum Fennicae Mathematica
  • 2020
We say a measure is C d-rectifiable if there is a countable union of C d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in R to be C d-rectifiable,
On the conformal dimension of product measures
Given a compact set E⊂Rd−1 , d⩾1 , write KE:=[0,1]×E⊂Rd . A theorem of Bishop and Tyson states that any set of the form KE is minimal for conformal dimension: If (X,d) is a metric space and
Necessary Condition for Rectifiability Involving Wasserstein Distance W2
A Radon measure μ is n-rectifiable if μ ≪ H n and μ-almost all of supp μ can be covered by Lipschitz images of R. In this paper we give a necessary condition for rectifiability in terms of the
Rigidity of Derivations in the Plane and in Metric Measure Spaces
Following Weaver we study generalized differential operators, called (metric) derivations, and their linear algebraic properties. In particular, for k = 1, 2 we show that measures on k-dimensional
Rectifiability of planes and Alberti representations
We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let $X$ be a pointwise doubling metric measure space. Let
Characterization of rectifiable measures in terms of 𝛼-numbers
We characterize Radon measures $\mu$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz
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 Euclidean


Hausdorff dimension and doubling measures on metric spaces
Vol′berg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures
Subsets of rectifiable curves in Hilbert space-the analyst’s TSP
We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do
Analysis of and on uniformly rectifiable sets
The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular
Probability Theory, an Analytic View
This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It
Stroock, Probability theory, an analytic view
  • 1993
The geometry of fractal sets. Cambridge Tracts in Mathematics
  • The geometry of fractal sets. Cambridge Tracts in Mathematics
  • 1986
Trois notes sur les ensembles parfaits linéaires
  • Enseignement Math
  • 1969