Fractional divisibility of spheres with partially generic sets of rotations
Volume 274 / 2026
Fundamenta Mathematicae 274 (2026), 175-185
MSC: Primary 28A05; Secondary 03E15, 33C55, 43A90, 57M60
DOI: 10.4064/fm250401-28-4
Published online: 22 July 2026
Abstract
We say that an $r$-tuple $(\gamma _1,\ldots ,\gamma _r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $\mathbb {S}^{d-1}$ if there is a non-constant function $f\in L^2(\mathbb {S}^{d-1})$ such that its translations by $\gamma _1,\ldots ,\gamma _r$ sum to the constant-1 function. Our main result shows, informally speaking, that fractional divisibility is impossible if at least $r/2$ rotations are “generic”.
