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  "body": "\n## Problem\n\nFor real numbers $\\lambda, \\alpha, \\theta$, let $H_{\\lambda,\\alpha,\\theta}$ be the [[almost Mathieu operator|D08SKTH2T2]] on $\\ell^2(\\Z)$.\nA Cantor set in $\\R$ is a nonempty compact set with no isolated points and with empty interior.\nIs the [[spectrum|BMKATJ802A]] of $H_{\\lambda,\\alpha,\\theta}$ a Cantor set for every irrational $\\alpha$, every $\\lambda \\ne 0$ and every $\\theta$?\n\n## Remarks\n\nThe answer is yes.\nAvila and Jitomirskaya proved it in a paper titled \"The Ten Martini Problem\", published in the Annals of Mathematics in 2009.\nBefore that, Bellissard and Simon showed that the spectrum is a Cantor set for a generic set of pairs $(\\lambda, \\alpha)$.\nThe name comes from Mark Kac, who offered ten martinis for a solution; Barry Simon gave the problem this name in his list of problems on almost periodic Schrödinger operators.\n\n## Explanations\n\n### L1 human\n\nA Cantor set is like the [[middle-thirds Cantor set|0BXNDZFQME]]: a closed set full of gaps at every scale, with no intervals inside it and no isolated points.\nThe problem asks whether the possible energies of an electron in a quasi-periodic chain always form such a set.\nIts solution shows that quasi-periodicity, however weak, breaks the spectrum into infinitely many pieces at every scale.\n",
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