Asymptotic density in quasi-logarithmic additive number systems

Abstract

We show that in quasi-logarithmic additive number systems $\mycal{A}$ all partition sets have asymptotic density, and we obtain a corresponding monadic second-order limit law for adequate classes of relational structures. Our conditions on the local counting function p(n) of the set of irreducible elements of $\mycal{A}$ allow situations which are not covered by the density theorems of Compton [6] and Woods [15]. We also give conditions on p(n) which are sufficient to show the assumptions of Compton's result are satisfied, but which are not necessarily implied by those of Bell and Burris [2], Granovsky and Stark [8] or Stark [14]