Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices

In this paper, we prove Hölder continuity of the integrated density of states for discrete quasiperiodic Jacobi $d\times d$ block matrices with Diophantine frequencies. The Hölder exponent is shown to be any $\beta$ such that $0<\beta <1\mathrm{/}(2{\kappa }^d)$, where ${\kappa }^d$ is the acceleration, i.e., the slope of the sum of the top $d$ Lyapunov exponents in the imaginary direction of the phase. This generalizes the Hölder continuity results in the Schrödinger operator setting in \cites{GS2,HS1}, and also strengthens them in that setting by covering more Diophantine frequencies. The proof is built on a new scheme for obtaining a local zero count for finite-volume characteristic polynomials from a global one.