Densities in disjoint unions (Q2702801)

The author proves necesssary and sufficient conditions satisfied by the lower and upper asymptotic density of three sets \(A, B\) and \(C = A \cup B\), of positive integers. These conditions are two inequalities. Proofs are constructive, and use the properties of piecewiese linear continuous functions. In the first part of the proof three functions \(a, b, c\) are constructed. In the second part using these three functions there are constructed sets \(A,B,C\) with prescribed lower and upper asymptotic density.