# Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces

## TL;DR

This paper extends volume bounds for singular strata from Ricci limit spaces to non-collapsed RCD(K,N) metric measure spaces, using a quantitative differentiation approach.

## Contribution

It generalizes existing volume bounds to a broader class of metric measure spaces, providing new estimates for singular strata and boundary enlargements.

## Key findings

- Volume bounds for singular strata in non-collapsed RCD spaces
- Volume estimate for boundary enlargements of ncRCD spaces
- Extension of Cheeger-Naber bounds to RCD setting

## Abstract

The aim of this note is to generalize to the class of non collapsed RCD(K,N) metric measure spaces the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in \cite{CheegerNaber13a}. The proof, which is based on a quantitative differentiation argument, closely follows the original one. As a simple outcome we provide a volume estimate for the enlargement of Gigli-DePhilippis' boundary (\cite[Remark 3.8]{DePhilippisGigli18}) of ncRCD(K,N) spaces.