Davenport and Erdös  proved that every set A of integers with the property that a ∈ A implies an ∈ A for all n (multiplicative ideal) has a logarithmic density. I generalized  this result to sets with the property that if for some numbers a, b, n we have a ∈ A, b ∈ A and an ∈ A, then necessarily bn ∈ A, which I call quasi-ideals. Here a new proof of this theorem is given, applying a result on convolution of measures on discretes semigroups. This leads to further generalizations, including an improvement of a result of Warlimont  on ideals in abstract arithmetic semigroups.
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