MULTIPLICATIVE STRUCTURE IN STABLE EXPANSIONS OF THE GROUP OF INTEGERS

摘要

We define two families of expansions of (Z,+) by unary predicates, and prove that their theories are superstable of U-rank ω. The first family consists of expansions (Z,+,A), where A is an infinite subset of a finitely generated multiplicative submonoid of Z~+. Using this result, we also prove stability for the expansion of (Z, +) by all unary predicates of the form {q~n : n ∈ N} for some q ∈ N_(>2). The second family consists of sets A ⊂ N which grow asymptotically close to a Q-linearly independent increasing sequence (λ_n)_(n=0)~∞⊂R~+ such that {(λ_n)/(λ_m):m≤n} is closed and discrete.