Fractional Divisibility of Spheres with Partially Generic Sets of Rotations

TL;DR

This paper investigates conditions under which certain rotations can fractionally divide spheres, showing that such divisibility is impossible when a majority of the rotations are generic.

Contribution

It introduces the concept of fractional divisibility of spheres by rotations and proves a new impossibility result for generic rotation sets.

Findings

Fractional divisibility is impossible with at least half of the rotations being generic.

Defines fractional divisibility in terms of functions on spheres and rotations.

Provides conditions under which fractional divisibility cannot occur.

Abstract

We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that its translations by $g_1,...,g_r$ sum up to the constant-1 function. Our main result shows, informally speaking, that fractional divisibility is impossible if at least $r/2$ rotations are ``generic".