Last's periodic approximation conjecture for almost periodic Schrödinger operators

Let $f$ be a real-valued analytic potential, let $H_{\beta,\theta}$ be the associated discrete Schrödinger operator, and for irrational $\beta$ let

$$\mathcal{A}(\beta)=\left\{E\mid \bar\gamma(E,\beta)=0\right\}.$$

For rational $p/q$, let $\sigma(p/q,\theta)$ denote the spectrum of the corresponding periodic operator, and set

$$S_-(p/q)=\bigcap_{0\leq\theta<2\pi}\sigma(p/q,\theta).$$

Last's periodic approximation conjecture. For any $\alpha\notin\mathbb{Q}$, $\mathcal{A}(\alpha)=\lim_{p/q\to\alpha}S_-(p/q)$, in the sense that

$$\limsup_{p/q\to\alpha}S_-(p/q)=\bigcap_{\delta>0}\bigcup_{\left|p/q-\alpha\right|<\delta}S_-(p/q)$$

and

$$\liminf_{p/q\to\alpha}S_-(p/q)=\bigcup_{\delta>0}\bigcap_{\left|p/q-\alpha\right|<\delta}S_-(p/q)$$

coincide, at least up to Lebesgue measure zero, with one another and with $\mathcal{A}(\alpha)$. This conjecture proposes that the zero-Lyapunov-exponent set for an irrational frequency is captured by the common spectral portions of nearby periodic approximants. The source attributes the conjecture to Y. Last; no resolution is supplied here.