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

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