Almost sure dimensional properties for the spectrum and the density of states of Sturmian Hamiltonians

In this paper, we find a full Lebesgue measure set of frequencies $\check{\text{II}}\subset [0,1]\setminus \text{Q}$ such that for any $(\alpha ,\lambda )\in \check{\text{II}}\times [24,\mathrm{\infty })$, the Hausdorff and box dimensions of the spectrum of the Sturmian Hamiltonian $H_{\alpha ,\lambda ,\theta }$ coincide and are independent of $\alpha$. Denote the common value by $D(\lambda )$, we show that $D(\lambda )$ satisfies a Bowen type formula, and is locally Lipschitz. We obtain the exact asymptotic behavior of $D(\lambda )$ as $\lambda$ tends to $\mathrm{\infty }\mathrm{.}$ This considerably improves the result of Damanik and Gorodetski (Comm. Math. Phys. 337, 2015). We also show that for any $(\alpha ,\lambda )\in \check{\text{II}}\times [24,\mathrm{\infty })$, the density of states measure of $H_{\alpha ,\lambda ,\theta }$ is exact-dimensional; its Hausdorff and packing dimensions coincide and are independent of $\alpha$. Denote the common value by $d(\lambda )$, we show that $d(\lambda )$ satisfies a Young type formula, and is Lipschitz. We obtain the exact asymptotic behavior of $d(\lambda )$ as $\lambda$ tends to $\mathrm{\infty }\mathrm{.}$ During the course of study, we also answer several questions in the same paper of Damanik and Gorodetski.