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 S + T = {m + n | m ∈ S, n ∈ T}, union and intersection, as well as ultimately periodic constants. An equation with a non-periodic solution of exponential growth is constructed. At the same time it is demonstrated that no sets with super-exponential growth can be represented. It is also shown that a restricted class of these equations cannot represent sets with super-linearly growing complements. The results have direct implications on the power of conjunctive grammars with one nonterminal