The Dry Ten Martini Problem for the critical almost Mathieu operator (remaining case: frequencies with $\beta(\alpha)=0$)

This task concerns the remaining case of the Dry Ten Martini Problem for the critical ($\lambda=1$) almost Mathieu operator, addressed in Borgnia–Slager's preprint "The Dry Ten Martini Problem at Criticality" (arXiv:2112.06869), whose constructive argument left three directions open: non-Diophantine frequencies, quantitative convergence control, and generalization beyond the single critical point. At critical coupling, all gap-labeled spectral gaps are already known to be open for every irrational frequency $\alpha$ with $0<\beta(\alpha)<\infty$ or $\beta(\alpha)=\infty$ (Avila–Jitomirskaya 2009, optimizing Choi–Elliott–Yui 1990) — the exponentially Liouville part of the preprint's proposed 'Liouville extension' is therefore already resolved at the level of the underlying statement — while the non-critical case $\lambda\neq\pm 1$ is completely solved for all irrational $\alpha$ (Avila–You–Zhou 2023). The surviving open core is the last case of the famous Dry Ten Martini Problem: prove that every spectral gap allowed by the gap-labeling theorem is open for the critical almost Mathieu operator at frequencies with $\beta(\alpha)=0$, a class that includes all Diophantine frequencies such as the golden mean. No proof that all gaps allowed by the gap-labeling theorem are open exists even at a single Diophantine frequency: Figueras–Puig (2026) computer-validated only finitely many individual gaps — eight for the golden mean and twelve for $\alpha=e-2$. The original 2021 preprint (v1) is not accepted by the mathematical literature as a resolution, and its own author now acknowledges that its argument did not establish the claimed theorem; a September 2026 revision (v2, Borgnia–Faust) claims a complete proof for every irrational frequency via substantially different methods, but it is non-refereed and has not been vetted by the community. Absent a verified resolution, the problem remains open.

Background

The almost Mathieu operator (AMO), also called the Aubry–André–Harper model in physics, acts on sequences $u\in\ell^2(\mathbb{Z})$ by

$$[H_{\lambda,\alpha,\theta}u](n)=u(n+1)+u(n-1)+2\lambda\cos(2\pi(n\alpha+\theta))u(n),$$

where $\theta$ is a phase, $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ is the (irrational) frequency of the quasiperiodic potential, and $\lambda\neq 0$ is the coupling constant. It models an electron on a two-dimensional lattice in a perpendicular magnetic field (after reduction along one direction), and its spectrum versus $\alpha$ is the Hofstadter butterfly. The parameter $\lambda=1$ is the critical coupling: there the operator is self-dual under Aubry duality (a Fourier-type transformation mapping $H_{\lambda,\alpha,\theta}$ to $H_{1/\lambda,\alpha,\theta'}$), and the spectrum is a zero-measure Cantor set with purely singularly continuous spectral measures.

For irrational $\alpha$ the spectrum $\Sigma_{\lambda,\alpha}$ is independent of $\theta$. Its complement consists of open intervals called spectral gaps. The integrated density of states $N_{\lambda,\alpha}(E)$ — the $\theta$-average of the spectral distribution — is continuous, strictly increasing on the spectrum, and constant on each gap; the gap-labeling theorem (Bellissard's noncommutative-geometry formulation of the Thouless charge pump) states that in any gap $N_{\lambda,\alpha}$ takes a value $k\alpha\bmod 1$ for a unique nonzero integer $k$, physically the Chern number (an integer topological invariant of a quantum Hall system) labeling that gap. A label $k$ is realized only if the corresponding gap is actually a nonempty interval; otherwise one speaks of a collapsed (closed) gap. The Ten Martini Problem — asked by Mark Kac in 1981 and popularized by Barry Simon — asks whether the spectrum is a Cantor set, equivalently whether open gaps correspond to a dense set of labels; it was solved affirmatively by Avila and Jitomirskaya (2009), building on Choi–Elliott–Yui (1990), Puig (2004), and others. The stronger Dry Ten Martini Problem asks whether every gap allowed by the gap-labeling theorem is open, i.e. no labeled gap ever collapses.

Progress on the Dry Ten Martini Problem splits by arithmetic properties of $\alpha$, quantified by its continued-fraction convergents $p_n/q_n$: $\alpha$ is Diophantine if $|\alpha-p_n/q_n|$ decays at worst polynomially in $q_n$ (badly approximable by rationals), and Liouville otherwise. The relevant exponent is $\beta(\alpha)=\limsup_n \ln q_{n+1}/q_n$, which measures how fast the denominators of the convergents grow: Diophantine (and more generally subexponentially approximated) frequencies have $\beta(\alpha)=0$, while $\beta(\alpha)>0$ or $\beta(\alpha)=\infty$ means $\alpha$ is well approximated by rationals at an exponential or superexponential rate. Reducibility methods — conjugating the eigenvalue transfer cocycle (the first-order rewriting of the eigenvalue equation as a skew-product dynamical system) to a constant cocycle, then applying the Moser–Pöschel perturbative gap-opening argument — settled the problem for $\beta(\alpha)=0$ but only at non-critical couplings $|\lambda|\neq 1$, while the C*-algebraic exponential estimates of Choi–Elliott–Yui, optimized by Avila–Jitomirskaya, settled it whenever $e^{-\beta(\alpha)}<\lambda<e^{\beta(\alpha)}$ (and trivially for $\beta(\alpha)=\infty$). In particular, at the critical coupling $\lambda=1$ the latter covers every frequency with $0<\beta(\alpha)<\infty$ or $\beta(\alpha)=\infty$ — that is, the frequencies approximated by rationals at an exponential rate or faster, a strict subclass of the Liouville numbers (Liouville numbers approximated at a merely subexponential rate have $\beta(\alpha)=0$ and lie on the open side). Avila, You, and Zhou (2023) then completed the entire non-critical case: all labeled gaps are open for every irrational $\alpha$ and every $\lambda\neq\pm 1$, via quantitative almost reducibility, a quantitative Aubry duality, and a generalized Moser–Pöschel argument. They explicitly exclude the critical couplings $\lambda=\pm 1$; the sign is immaterial to the spectrum, since $\Sigma_{-\lambda,\alpha}=\Sigma_{\lambda,\alpha}$ by the phase shift $\theta\mapsto\theta+1/2$.

What remains is criticality itself for $\beta(\alpha)=0$ frequencies. There every existing mechanism degenerates: the Lyapunov exponent (the asymptotic exponential growth rate of the transfer matrices, which drives hyperbolicity-based gap opening) vanishes at $\lambda=1$, the cocycle is neither uniformly hyperbolic nor reducible, and Aubry duality maps the critical operator to itself so the non-critical duality bootstrap cannot run. The surrounding critical structure is well mapped — the spectrum has zero Lebesgue measure, Hausdorff dimension at most $1/2$ with gap continuity (Jitomirskaya–Krasovsky 2019), no point spectrum for any irrational $\alpha$ (Jitomirskaya 2021) — but none of this determines which gaps are open.

The unpublished preprint of Borgnia and Slager (2021) approaches exactly this critical case from condensed-matter physics: a chiral gauge transformation (a phase redefinition of the hopping amplitudes that makes the critical AMO the isotropic Hofstadter Hamiltonian on a longer unit cell), transfer matrices built from rational continued-fraction approximants $p_N/q_N$, and projected Green's functions (spectral projections onto one edge of a half-space, whose zeros detect gap-edge states), combined with bulk-boundary correspondence (the principle that the edge modes of a half-space system encode the bulk topological invariants) to place a zero of the projected Green's function inside every Chern-labeled gap. Their convergence argument for the irrational limit is restricted to Diophantine $\alpha$ and passes through complex energies $\omega\pm i\epsilon$ with a two-limit procedure $\lim_{\epsilon\to 0}\lim_{q_N\to\infty}$; the preprint was never published, and the rigorous literature does not treat it as settling the critical case — a September 2026 revision of the same arXiv entry (v2, Borgnia–Faust) acknowledges that this original argument did not establish the claimed theorem and instead claims a complete proof via substantially different methods. The most recent peer-reviewed mathematical state of the art is the computer-assisted validation of Figueras and Puig (2026), which certifies, by constructive conjugation to a hyperbolic cocycle and interval-arithmetic eigenvalue bounds for periodic approximants, only finitely many individual gaps: eight (the bounded gaps with labels $|k|\leq 4$) for the golden mean and twelve for $\alpha=e-2$. Their paper states plainly that whether all possible gaps are open for the AMO 'remains open only in the critical case', and that for a Diophantine frequency such as the golden mean it was not even known whether any particular gap is open.

Problem Statement

Prove or refute the last open case of the Dry Ten Martini Problem for the almost Mathieu operator. For the critical operator $H_{1,\alpha,\theta}$ ($\lambda=1$, the self-dual Aubry–André–Harper model), show that for every irrational frequency $\alpha$ with $\beta(\alpha)=\limsup_n\ln q_{n+1}/q_n=0$ — in particular for every Diophantine $\alpha$, such as the golden mean $(\sqrt{5}-1)/2$ — every spectral gap allowed by the gap-labeling theorem is open: for each $k\in\mathbb{Z}\setminus\{0\}$ there is a nonempty open interval $G\subset\mathbb{R}\setminus\Sigma_{1,\alpha}$ (an open spectral gap) on which the integrated density of states $N_{1,\alpha}$ is constant with value $k\alpha\bmod 1$, where $\Sigma_{1,\alpha}$ is the spectrum (independent of the phase $\theta$) and $q_n$ are the denominators of the continued-fraction convergents of $\alpha$. Equivalently, given the prior results for $\lambda\neq\pm 1$ (all irrational $\alpha$) and for $\lambda=\pm 1$ with $\beta(\alpha)>0$ or $\beta(\alpha)=\infty$: all gap-labeled gaps of the almost Mathieu operator are open for all irrational $\alpha$ and all $\lambda\neq 0$. A refutation would consist of an explicit irrational $\alpha$ with $\beta(\alpha)=0$ and a gap label $k$ for which the $k$-th gap is collapsed at $\lambda=1$ (the two gap boundaries coincide). Any rigorous method is admissible.

References

1. Dan S. Borgnia and Robert-Jan Slager, "The Dry Ten Martini Problem at Criticality", arXiv:2112.06869 (2021); revised as v2 (2026) by Dan S. Borgnia and Matthew Faust, claiming a complete proof via substantially different methods. https://arxiv.org/abs/2112.06869
2. Artur Avila and Svetlana Jitomirskaya, "The Ten Martini Problem", Annals of Mathematics 170 (2009) 303–342. DOI: 10.4007/annals.2009.170.303, https://doi.org/10.4007/annals.2009.170.303
3. Man-Duen Choi, George A. Elliott, and Noriko Yui, "Gauss polynomials and the rotation algebra", Inventiones Mathematicae 99 (1990) 225–246. DOI: 10.1007/BF01234419, https://doi.org/10.1007/BF01234419
4. Joaquim Puig, "Cantor Spectrum for the Almost Mathieu Operator", Communications in Mathematical Physics 244 (2004) 297–309. DOI: 10.1007/s00220-003-0977-3, https://doi.org/10.1007/s00220-003-0977-3
5. Svetlana Jitomirskaya, "On point spectrum of critical almost Mathieu operators", Advances in Mathematics 392 (2021) 107997. DOI: 10.1016/j.aim.2021.107997, https://doi.org/10.1016/j.aim.2021.107997
6. Svetlana Jitomirskaya and Igor Krasovsky, "Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum", arXiv:1909.04429 (2019). https://arxiv.org/abs/1909.04429
7. Artur Avila, Jiangong You, and Qi Zhou, "Dry Ten Martini Problem in the non-critical case", arXiv:2306.16254 (2023). https://arxiv.org/abs/2306.16254
8. Jordi-Lluís Figueras and Joaquim Puig, "Computer validation of open gaps for the almost Mathieu operator with critical coupling", Communications in Nonlinear Science and Numerical Simulation 152 (2026) 109407. DOI: 10.1016/j.cnsns.2025.109407, https://doi.org/10.1016/j.cnsns.2025.109407 (preprint arXiv:2410.18536)
9. Martin Leguil, Jiangong You, Zhiyan Zhao, and Qi Zhou, "Asymptotics of spectral gaps of quasi-periodic Schrödinger operators", arXiv:1712.04700 (2017). https://arxiv.org/abs/1712.04700
10. Ram Band, Siegfried Beckus, and Raphael Loewy, "The Dry Ten Martini Problem for Sturmian Hamiltonians", arXiv:2402.16703 (2024). https://arxiv.org/abs/2402.16703
11. Lingrui Ge, Yiqian Wang, and Jiahao Xu, "The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schrödinger operators", arXiv:2503.06918 (2025). https://arxiv.org/abs/2503.06918
12. Christopher Cedzich and Long Li, "Twenty dry Martinis for the Unitary Almost Mathieu Operator", arXiv:2503.06710 (2025). https://arxiv.org/abs/2503.06710
13. Xianzhe Li, Disheng Xu, and Qi Zhou, "Monotonicity, global symplectification and the stability of Dry Ten Martini Problem", arXiv:2601.02222 (2026). https://arxiv.org/abs/2601.02222