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

@inproceedings{Garnett2009ADM, title={A doubling measure on R^d can charge a rectifiable curve}, author={John B. Garnett and Rowan Killip and Raanan Schul}, year={2009} }

For d > 2, we construct a doubling measure v on ℝ d and a rectifiable curve Γ such that ν(Γ) > 0.

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## References

SHOWING 1-8 OF 8 REFERENCES

Hausdorff dimension and doubling measures on metric spaces

- Mathematics
- 1998

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

- Mathematics
- 2006

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

- Mathematics
- 1993

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

- Mathematics
- 1993

PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular…

Probability Theory, an Analytic View

- Mathematics
- 1993

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