Stability and sparsity in sets of natural numbers

  • Gabriel ConantEmail author


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 supa∈A |af(a)| < ∞ and {\(\frac{s}{t}:s,t \in f(A) \), ts} 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.


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© Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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