Fractional Divisibility of Spheres with Partially Generic Sets of Rotations

We say that an r-tuple $(g_1,\mathrm{.}\mathrm{.}\mathrm{.},g_r)$ 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 $g_1,\mathrm{.}\mathrm{.}\mathrm{.},g_r$ sum up to the constant-1 function. Our main result shows, informally speaking, that fractional divisibility is impossible if at least $r\mathrm{/}2$ rotations are ``generic".