On the structure of universal differentiability sets

TL;DR

This paper investigates the structural properties of universal differentiability sets in Euclidean spaces, revealing that they contain dense 'kernels' and cannot be decomposed into countable unions of non-universal sets, highlighting their unique complexity.

Contribution

It establishes new structural properties of universal differentiability sets, including the existence of dense kernels and indecomposability into certain unions, advancing understanding of their geometric nature.

Findings

Universal differentiability sets contain a dense 'kernel' of differentiability points.

Such sets cannot be expressed as a countable union of non-universal differentiability sets.

The results are sharp compared to existing decomposability theorems.

Abstract

We prove that universal differentiability sets in Euclidean spaces possess distinctive structural properties. Namely, we show that any universal differentiability set contains a `kernel' in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets. The sharpness of this result, with respect to existing decomposibility results of the opposite nature, is discussed.