For two inner functions $\vartheta,\varphi\in H^\infty$, we give a simple sufficient condition for the system $\vartheta^m,\; \varphi^n$, $m,n\in\mathbb{ Z}$, to be complete in the weak-$^*$ topology of $L^\infty(\mathbb{ T})$. To be precise, we show that this system is complete whenever there is an arc $I$ of the unit circle $\mathbb{ T}$ such that $\vartheta$ is univalent on $I$ and $\varphi$ is univalent on $\mathbb{ T}\setminus I$. As an application of this result, we describe a class of analytic curves $\Gamma$ such that $(\Gamma, \mathcal{ X})$ is a Heisenberg uniqueness pair, where $\mathcal{ X}$ is the lattice cross $\{ (m,n)\in\mathbb{ Z}^2:\, mn=0\}$. Our main result extends a theorem of Hedenmalm and Montes-Rodr\'iguez for atomic inner functions with one singularity.
Categories: math.FA math.CV