Proposition: Exact criterion for absence of a strict diffractive coefficient

The function ${\mathcal{A}}_{\alpha ,\beta }$ vanishes identically on a nonempty transition-free angular interval if and only if $(\alpha ,\beta )\in \mathcal{Q}$.

If $(\alpha ,\beta )\notin \mathcal{Q}$, its nonzero angular values are dense in the complement of ${\mathcal{B}}_{\alpha }$, and

Proof. Multiplying the numerator in (eq:A) by $e^{-\mathrm{i}\beta \delta }$ gives

It vanishes on an interval precisely when both constants vanish:

Their sum gives $1/\alpha =N\in \mathbb{N}$, $N\ge 2$, and then $\beta =p/N$ with $0\le p<N$. The converse is immediate.

Outside this exceptional case, the analytic numerator has no open zero interval. At every nonzero coefficient value, Lemmas lem:diffnormal and lem:sqrtWF give all of ${\mathcal{D}}_{\mathrm{ap}}^{+}$ over that angular value. Closedness of the joint wavefront set supplies the same radial covectors at its angular zeros and at the transition angles. This proves (eq:difflower).