On the expressive power of univariate equations over sets of natural numbers

Equations of the form X=φ(X) are considered, where the unknown X is a set of natural numbers. The expression φ(X) may contain the operations of set addition, defined as , union, intersection, as well as ultimately periodic constants. An equation with a non-periodic solution of exponential growth rate is constructed. At the same time it is demonstrated that no sets with super-exponential growth rate can be represented. It is also shown that restricted classes of these equations cannot represent sets with super-linearly growing complements nor sets that are additive bases of order 2. The results have direct implications on the power of unary conjunctive grammars with one nonterminal symbol.