Local metric properties and regular stratifications of p-adic definable sets

TL;DR

This paper investigates the local geometric and measure-theoretic properties of p-adic definable sets, establishing the existence of local density, introducing distinguished tangent cones, and deriving a p-adic version of the Cauchy-Crofton formula.

Contribution

It introduces the notion of distinguished tangent cones in the p-adic setting and proves their stabilization, extending real geometric concepts to p-adic definable sets.

Findings

Local density of p-adic definable sets exists at each point.

Distinguished tangent cones can be used to compute local density up to multiplicities.

A p-adic analogue of the Cauchy-Crofton formula for density is established.

Abstract

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a $p$-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our field and show, as it is the case in the real setting, that, up to some multiplicities, the local density may be computed on this distinguished tangent cone.We also prove that these distinguished tangent cones stabilize for small enough subgroups. We finally obtain the $p$-adic counterpart of the Cauchy-Crofton formula for the density. To prove these results we use the Lipschitz decomposition of definable $p$-adic sets of arXiv:0904.3853v1 and prove here the genericity of the regularity conditions for stratification such as $(w_f)$, $(w)$, $(a_f)$, $(b)$ and $(a)$ conditions.