Quantitative Reifenberg theorem for measures

Abstract We study generalizations of Reifenberg’s Theorem for measures in $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> under assumptions on the Jones’ $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which hold for general measures without density assumptions, give effective measure bounds on $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> 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 $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> . Applications include quantitative Reifenberg theorems on sets and discrete measures, as well as upper Ahlfor’s regularity estimates on measures which satisfy $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -number estimates on all scales.