It is known that there exist classes of sets of positive integers which provide arbitrary given values as lower and upper logarithmic and asymptotic densities. The only restriction is the inequality between these densities. We generalize this result for a sequence of weighted densities, i.e., for countably infinitely many weighted densities.
"Sets with countably infinitely many prescribed weighted densities." Rocky Mountain J. Math. 50 (2) 467 - 477, April 2020. https://doi.org/10.1216/rmj.2020.50.467