Polynomial Spectral Gap Estimates for Finitely Differentiable Quasi-Periodic Schrödinger Operators

We prove polynomial upper bounds on spectral gaps of quasi-periodic Schrödinger operators with small $C^{k}$ potentials and Diophantine frequencies. For sufficiently large $k$, the decay exponent is $k-34τ$, where $τ$ is the Diophantine exponent. We construct examples with lower bounds along infinit