Dispersive estimates for discrete Klein-Gordon equations on one-dimensional lattice with quasi-periodic potentials

We prove ${\mathrm{\ell }}^1\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\mathrm{\ell }}^{\mathrm{\infty }}$ dispersive estimates for the discrete Klein–Gordon equation on $\mathbb{Z}$ with small real-analytic quasi-periodic potentials, showing that the time-decay rate persists as $(\frac{1}{3}{)}^{-}$. As applications, we derive the corresponding Strichartz estimates and establish small-data global well-posedness for the associated nonlinear discrete Klein–Gordon equation.