On Sets of Singular Rotations for Translation Invariant Bases

TL;DR

This paper investigates the set of rotations that cause non-differentiability of integrals with respect to translation-invariant bases, revealing their topological structure for the plane.

Contribution

It characterizes the topological structure of singular rotation sets for translation-invariant bases on the plane, advancing understanding of differentiability in geometric measure theory.

Findings

Identifies the possible topological structures of singular rotation sets

Provides a classification for translation-invariant bases on the plane

Enhances understanding of differentiability with respect to rotations

Abstract

It is studied the following problem: for a given function $f$ what kind of may be a set of all rotations $\gamma$ for which $\int f$ is not differentiable with respect to $\gamma$-rotation of a given basis $B$? In particular, for translation invariant bases on the plane it is found the topological structure of possible sets of singular rotations.