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We study the Anderson orthogonality catastrophe, and the corresponding x-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-André model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wave functions and the density of states are fractal. We find, numerically, that the overlap between postquench and prequench wave functions, as well as the “core-hole” Green function, evolve in a complex, nonmonotonic way with system size and time, respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the postquench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically …