Motivic Vitushkin invariants

TL;DR

This paper establishes a nonarchimedean analogue of a real inequality linking metric entropy and geometric invariants, using motivic integration and introducing new motivic variants of Vitushkin's variations.

Contribution

It introduces nonarchimedean versions of Vitushkin variations and proves a global Cauchy-Crofton formula within the motivic setting, extending prior real inequalities.

Findings

Proves a nonarchimedean inequality involving motivic invariants.

Defines motivic counterparts of Vitushkin's variations, including $V_0$.

Establishes a nonarchimedean Cauchy-Crofton formula for definable sets.

Abstract

We prove the nonarchimedean counterpart of a real inequality involving the metric entropy and measure geometric invariants $V_i$, called Vitushkin's variations. Our inequality is based on a new convenient partial preorder on the set of constructible motivic functions, extending the one considered by R. Cluckers and F. Loeser in Constructible motivic functions and motivic integration, Invent. Math., 173 (2008). We introduce, using motivic integration theory and the notion of riso-triviality, nonarchimedean substitutes of the Vitushkin variations $V_i$, and in particular of the number $V_0$ of connected components. We also prove the nonarchimedean global Cauchy-Crofton formula for definable sets of dimension $d$, relating $V_d$ and the motivic measure in dimension $d$.