Logarithmic density and measures on semigroups

Davenport and Erdős 3] proved that every setA of integers with the property thata ∈A impliesan ∈A for alln (multiplicative ideal) has a logarithmic density. I generalized 5] this result to sets with the property that if for some numbersa, b, n we havea ∈ A, b ∈ A andan ∈ A, then necessarilybn ∈ 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 8] on ideals in abstract arithmetic semigroups. Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901