Corollary

Let C be the open angular sector formed by the union of the open half-lines $\Delta_1$ starting at the origin and such that $0 < (\widehat{\Delta}, \widehat{\Delta_1}) < \theta$. Then no transform of D or D' by an element of W meets C if and only if m is finite and

If $m = \infty$, the image of the set $n \theta$ ($n \in \mathbf{Z}$) is dense in $\mathbf{R}/2\pi \mathbf{Z}$ (General Topology, Chap. VII, § 1, Prop. 11); the union of the transforms of D by the elements of W is thus dense in V and meets C. If m is finite and if $\theta = k \pi / m$ with $1 < k < m$, the integers k and m being relatively prime, there exists an integer h such that $hk \equiv 1$ mod. m; then $(D, \rho_{h \theta}(D)) \equiv \pi / m$ (mod. $\pi$), and $\rho_{h \theta}(D)$ meets C. This shows that the condition is necessary. The converse is immediate.