The class of the exceptional sets for a general asymptotic formula

Abstract

We study the problem of the existence of a true exceptional set for an asymptotic formula, that is a minimal set --- up to finite modifications --- such that the asymptotic formula holds outside such a set. We give an analytic and a descriptive set theoretic characterisations for the existence of a true exceptional set, which we then apply by showing the non-existence of a true exceptional set in some well known situations. We prove in fact that, both from a category and a measure theoretic points of view, most asymptotic formulas do not have a true exceptional set.