Fractional Divisibility of Spheres with Partially Generic Sets of Rotations

Jan Grebík; Christian Ikenmeyer; Oleg Pikhurko

paper

Fractional Divisibility of Spheres with Partially Generic Sets of Rotations

arXiv:2501.13522

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".

13 pages, minor changes, accepted by Fundamenta Mathematicae, reformatted in journal's style