Density sets of sets of positive integers (Q977128)

A nonnegative function \(f\) on the set of positive integers induces a concept of upper density by \[ \overline d(A) = \limsup \Bigl( \sum _{a\leq n, a\in A} f(a) \Bigr) \Bigm/ \sum _{k=1}^n f(k), \] and a lower density analogously. Given a set \(A\), its density set defined by \[ S(A) = \{ ( \overline d(X), \underline d(X) ): X\subset A\} \subset [0,1]^2 . \] For a set which fails to have a density this contains more information than just the upper and lower densities. The authors describe this set for a wide class of functions, which include familiar examples like the asymptotic and logarithmic density.