"Are all gaps there?", asked Mark Kac in 1981 during a talk at the AMS annual meeting,
and offered ten Martinis for the answer.
This led Barry Simon to coin the names the Ten Martini Problem (TMP) and
 the Dry Ten Martini Problem for two related problems concerning the Almost-Mathieu operator.
The TMP is to show that the spectrum of the Almost-Mathieu operator is a Cantor set.
The Dry TMP concerns the values of the integrated density of states (IDS).
The gap labelling theorem predicts the possible set of values which the IDS may attain at the spectral gaps. The Dry TMP is whether or not all these values are attained, or equivalently, "are all gaps there?".
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The TMP was fully solved by Artur Avila and Svetlana Jitomirskaya in 2005.
The Dry TMP has been recently announced to be solved for the non-critical case (coupling constant different than one).
The corresponding preprint by Artur Avila, Jiangong You and Qi Zhou appeared on June 2023.
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This talk is about the Dry TMP for Sturmian Hamiltonians.
These are one-dimensional Schroedinger operators with aperiodic potentials determined by Sturmian sequences.
The potential is determined in terms of two parameters: the frequency and the potential strength (a.k.a coupling constant).
As for the Almost-Mathieu operator the Dry TMP is whether all the possible spectral gaps are there for all irrational frequencies and all coupling constants.
For large values of the coupling constant, the Sturmian Dry TMP was solved by Raymond in 1997.
In 2016, David Damanik, Anton Gorodetski and William Yessen provided a solution if the frequency is the golden mean and for all couplings.
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In a current project with Siegfried Beckus and Raphael Loewy we solve the Sturmian Dry TMP for all irrational frequencies and all couplings.
In the talk we present the problem and the route to its resolution.