Hausdorff Dimension of Critical Almost Mathieu Spectra

Definitions and mathematical statement

For real parameters $\alpha,\lambda,\theta$, define the bounded self-adjoint almost Mathieu operator on $\ell^2({\mathbb{Z}})$ by \[(H_{\alpha,\lambda,\theta}\psi)(n)=\psi(n+1)+\psi(n-1)+\lambda\cos(2\pi\alpha n+\theta)\psi(n).\] For irrational $\alpha$, let $S(\alpha,\lambda)\subseteq{\mathbb{R}}$ be its spectrum, which the source states is independent of $\theta$. The coupling normalization is exactly $\lambda\cos(\cdot)$; thus the specified critical coupling is $\lambda=2$, rather than the value one used with a $2\lambda\cos(\cdot)$ convention. Write $\dim_H A$ for Hausdorff dimension, the critical exponent for vanishing Hausdorff measure formed from covers weighted by powers of their diameters. The conjecture asks whether \[\forall\alpha\in{\mathbb{R}}\setminus{\mathbb{Q}}:\qquad\dim_H S(\alpha,2)<\tfrac12.\] The inequality is strict for each frequency. It is neither the non-strict bound $\leq\tfrac12$ nor a request for one positive dimension gap uniform in $\alpha$. The supplied 2026-08-21 triage labels the record partially solved, credits Last and Shamis with zero dimension on a dense $G_\delta$ set, and reports a later non-strict one-half upper bound for every irrational frequency. It says strict inequality and a full dimension classification remain open; these are supplied literature claims, not an independent verification.

Short English statement

At critical coupling two in the source's normalization, must every irrational-frequency almost Mathieu spectrum have Hausdorff dimension strictly less than one half? A reported universal non-strict bound does not answer this strict pointwise question.

Sources

• Local UnsolvedMath registry, permanent ID 30000942, source problem catalogue entry, Hausdorff Dimension of Critical Almost Mathieu Spectra, proposed year 2008; no individual proposer supplied.
• Supplied citation: Random Schrödinger Operators and Random Matrices (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/16, pp. 797--868: https://doi.org/10.4171/owr/2008/16. Supplied literature label: Oberwolfach Report 16/2008.
• Supplied literature label: Last and Shamis, zero Hausdorff dimension, https://arxiv.org/abs/1510.07651.
• Supplied literature label: critical almost Mathieu spectrum, https://arxiv.org/abs/1909.04429.