“Are all gaps there?” – a question originally posed by Mark Kac for the Almost Mathieu operator and later termed the Dry Ten Martini Problem by Simon – asks whether all gap labels predicted by the Gap Labelling Theorem are attained by the integrated density of states. For the Almost Mathieu operator and for Sturmian Hamiltonians, this problem has been resolved in many regimes, with all allowed gaps shown to be open.

In this talk, we introduce a geometric analogue of this problem for operators on quasiperiodic graphs generated by Sturmian dynamical systems. Here the aperiodicity is encoded not in a potential, but in the underlying geometric structure. We compute the corresponding gap labels explicitly and show that, in contrast to the classical Sturmian Hamiltonians, not all allowed gaps are open.

The missing gaps arise from a geometric mechanism: the local structure of the graph produces flat bands, leading to unattained gap labels. This demonstrates how global quasiperiodic structure and local geometry can interact to determine spectral structure. Based on joint work with Ram Band.