Fractional Divisibility of Spheres with Partially Generic Sets of Rotations
Jan Grebík, Christian Ikenmeyer, Oleg Pikhurko
TL;DR
The paper investigates when an $r$-tuple of rotations can fractionally divide the sphere $S^{d-1}$ under the action of $SO(d)$. By translating fractional divisibility into the invertibility of linear operators $L_n$ on spherical harmonic subspaces $H_n$, it leverages harmonic analysis and algebraic geometry to show that, for $d\ge 3$, fixing at least $\lfloor r/2\rfloor$ rotations forces non-divisibility for all configurations of the remaining rotations; the case $d=2$ yields a stronger countable-structure result. The main contributions are explicit bounds $f^{top}(d,r)\le \lfloor r/2\rfloor$ and $f^{Haar}(d,r)\le \lfloor r/2\rfloor$ (with $d=2$ giving $1$), along with careful lower-bound observations. The results connect Fourier-analytic structures on spheres with algebraic-geometry arguments about zero sets of determinant polynomials, clarifying when fractional tilings by rotations are impossible and enriching the study of translational/divisibility phenomena on spheres. The findings have potential implications for understanding non-tiling and partitioning constraints in higher-dimensional spherical settings.
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".