Sets with prescribed arithmetic densities

Using concepts of generalized asymptotic and logarithmic densities based on weighted arithmetic means over an arithmetical semigroup $\mathbb{G}$ we prove that under some additional technical assumptions on the weighted counting function of its elements, a subset of $\mathbb{G}$ exists with all four generalized densities (upper and lower asymptotic and logarithmic) prescribed subject to the natural condition $0 \leq \underline{d}(A) \leq \underline{\ell}(A) \leq \overline{\ell}(A) \leq \overline{d}(A) \leq1$.