# Rectifiability and approximate differentiability of higher order for sets

## TL;DR

This paper introduces a higher order approximate differentiability concept for subsets of Euclidean space, enabling characterization of higher order rectifiable sets and extending classical function theory to sets.

## Contribution

It develops a new notion of approximate differential of order k for sets, proving it is a Borel map, thus advancing the understanding of higher order rectifiability.

## Key findings

- Defines approximate differential of order k for sets
- Proves the differential is a Borel map
- Provides a framework for higher order rectifiable sets

## Abstract

The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset $ A $ of the Euclidean space and for every integer $ k \geq 2 $ we introduce the approximate differential of order $ k $ of $ A $ and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications.