On rectifiable measures in Carnot groups: Marstrand–Mattila rectifiability criterion

In this paper we continue the study of notion P -rectifiability in Carnot groups. We say that a Radon measure is h -rectifiable, for ? N , if it has positive -lower density and finite -upper almost everywhere, and, at every point, admits unique tangent up to multiples. prove Marstrand–Mattila rectifiability criterion arbitrary groups -rectifiable measures with planes admit normal complementary subgroup. Namely, co-normal case, even priori point might not be same different scales, posteriori everywhere. Since horizontal subgroup group complement, our applies particular case which tangents are one-dimensional subgroups. Hence, as an immediate consequence result Chousionis–Magnani–Tyson, obtain Preiss's theorem first Heisenberg H 1 . More precisely, show ? on one-density respect Koranyi distance absolutely continuous Hausdorff supported one-rectifiable set sense Federer, i.e., countable union images Lipschitz maps from A ? R