  Inspired by the theory of p-adic differential equations, this paper
introduces an analogous theory for q-difference equations over a local field,
when |q|=1. We define some basic concepts, for instance the generic radius of
convergence, introduce technical tools, such as a twisted Taylor formula for
analytic functions, and prove some fundamental statements, such as an effective
bound theorem, the existence of a weak Frobenius structure and a transfer
theorem in regular singular disks.
