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arXiv · 1612.08052

Quantitative Reifenberg theorem for measures

Nick Edelen·Aaron Naber·Daniele Valtorta
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Abstract

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $β$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds on $μ$ away from a closed $k$-rectifiable set with bounded Hausdorff measure. We show examples to see the sharpness of our results. Under further density assumptions one can translate this into a global measure bound and $k$-rectifiable structure for $μ$. Applications include quantitative Reifenberg theorems on sets and discrete measures, as well as upper Ahlfor's regularity estimates on measures which satisfy $β$-number estimates on all scales.

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Nick Edelen, Aaron Naber, Daniele Valtorta. 2025-03-24. Quantitative Reifenberg theorem for measures. https://arxiv.org/abs/1612.08052

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