Stability and sparsity in sets of natural numbers

Abstract

Given a set A ⊆ ℕ, we consider the relationship between stability of the structure (ℤ, + , 0,A) and sparsity of the set A. We first show that a strong enough sparsity assumption on A yields stability of (ℤ, +, 0, A). Specifically, if there is a function f: A → ℝ⁺ such that sup_a∈A |a − f(a)| < ∞ and {$\frac{s}{t}:s,t \in f(A)$, t ≤ s} is closed and discrete, then (ℤ, +, 0, A) is superstable (of U-rank ω if A is infinite). Such sets include examples considered by Palacín and Sklinos [19] and Poizat [23], many classical linear recurrence sequences (e.g., the Fibonaccci numbers), and any set in which the limit of ratios of consecutive elements diverges. Finally, we consider sparsity conclusions on sets A ⊆ N, which follow from model theoretic assumptions on (ℤ, +, 0, A). We use a result of Erdős, Nathanson and Sárközy [8] to show that if (ℤ, +, 0, A) does not define the ordering on ℤ, then the lower asymptotic density of any finitary sumset of A is zero. Finally, in a theorem communicated to us by Goldbring, we use a result of Jin [11] to show that if (ℤ, +, 0,A) is stable, then the upper Banach density of any finitary sumset of A is zero.