 
Summary: THE TEN MARTINI PROBLEM
ARTUR AVILA AND SVETLANA JITOMIRSKAYA
Abstract. We prove the conjecture (known as the "Ten Martini Problem" after Kac and Simon)
that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the
coupling and all irrational frequencies.
1. Introduction
The almost Mathieu operator is the Schr¨odinger operator on 2
(Z),
(1.1) (H,,u)n = un+1 + un1 + 2 cos2( + n)un,
where , , R are parameters (called the coupling, frequency, and phase, respectively), and one
assumes that = 0. The interest in this particular model is motivated both by its connections to
physics and by a remarkable richness of the related spectral theory. This has made the latter a
subject of intense research in the last three decades (see [L2] for a recent historical account and for
the physics background). Here we are concerned with the topological structure of the spectrum.
If = p
q is rational, it is well known that the spectrum consists of the union of q intervals called
bands, possibly touching at the endpoints. In the case of irrational , the spectrum , (which in
this case does not depend on ) has been conjectured for a long time to be a Cantor set (see a 1964
paper of Azbel [Az]). To prove this conjecture has been dubbed The Ten Martini Problem by Barry
Simon, after an offer of Mark Kac in 1981, see Problem 4 in [Sim1]. For a history of this problem
