3 problems

Let $p$ and $q$ be coprime integers with $q$ odd, and let $\Sigma_{p/q}$ denote the periodic almost Mathieu spectrum. Let $C=0.915\ldots$ be Catalan's constant. Thouless-type spect…

Let $p$ and $q$ be coprime integers with $q$ odd. For the periodic problem $\Sigma(p/q)$, let $\gamma_k(p/q)$ denote the length of the gap with label $k$. Gap-decay and extremal-fr…

Almost Mathieu diffusion conjecture. For almost every $\theta,\alpha\in[0,1]$, the equation has a diffusion constant $D(g,\lambda)\in(0,\infty)$ that is smooth for all…