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Abstract

We study the question of whether or not sets of natural numbers with a given multiplicative structure are guaranteed to have a density. We consider several different types of multiplicative structure (multiplicatively closed sets, sets of multiples, saturated sets, and multiplicatively closed saturated sets), and several different types of density (natural density, logarithmic density, and multiplicative density). Many of these cases have been studied before; in this paper, we finish off the problem, answering the above question in every case.