Let$X$be a rationally convex compact subset of the unit sphere$S$in ℂ^{2}, of three-dimensional measure zero. Denote by$R$($X$) the uniform closure on$X$of the space of functions$P$/$Q$, where$P$and$Q$are polynomials and$Q$≠0 on$X$. When does$R$($X$)=$C$($X$)?