Motivic local density

TL;DR

This paper develops a motivic framework for local densities and tangent cones, extending p-adic density theory, and establishes geometric and stratification results for definable sets in Henselian valued fields.

Contribution

It introduces a motivic local density concept, proves the existence of tangent cones with motivic multiplicities, and applies these to uniform p-adic density results.

Findings

Existence of regular stratifications satisfying Verdier conditions.

Definition of motivic local density in the Grothendieck ring.

Uniformity theorem for p-adic local density.

Abstract

We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers-Comte-Loeser about $p$-adic local density. We prove some results about geometry of definable sets in Henselian valued fields of characteristic zero, both in semi-algebraic and subanalytic languages, and study Lipschitz continuous maps between such sets. We prove existence of regular stratifications satisfying analogous of Verdier condition $(w_f)$. Using Cluckers-Loeser theory of motivic integration, we define a notion of motivic local density with values in the Grothendieck ring of the theory of the residue sorts. We then prove the existence of a distinguished tangent cone and that one can compute the local density on this cone endowed with appropriate motivic multiplicities. As an application we prove a uniformity theorem for $p$-adic local density.