We study large deviations for measurable averaging operators on state spaces of dynamical systems. We prove a relatively sharp large deviation result in terms of the norm gap of the averaging operator. Developing ideas of Linnik and Ellenberg, Michel, and Venkatesh, we deduce an effective equidistribution theorem. The novelty of our results is that they apply to measures with suboptimal bounds on the mass of Bowen balls. We present two new applications of our results. The first one is effective rigidity for the measure of maximal entropy on S-arithmetic quotients. This is a partial extension of a recent result of Rühr. The second one is non-escape of mass for measures having large entropy in a non-Archimedean place. This generalizes known results for real flows.
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