Stability and sparsity in sets of natural numbers

Gabriel Conant

Published 2017-01-05Version 1

Given a set $A\subseteq\mathbb{N}$, we consider the relationship between stability of the structure $(\mathbb{Z},+,0,A)$ and sparsity assumptions on the set $A$. We first show that a strong enough sparsity assumption on $A$ yields stability of $(\mathbb{Z},+,0,A)$. Specifically, if there is a function $f:A\longrightarrow\mathbb{R}^+$ such that $\text{sup}_{a\in A}|a-f(a)|<\infty$ and $\{\frac{s}{t}:s,t\in f(A),~t\leq s\}$ is closed and discrete, then $(\mathbb{Z},+,0,A)$ is superstable (of $U$-rank $\omega$ if $A$ is infinite). Such sets encompass many classical examples of linear recurrence sequences (e.g. the Fibonaccci numbers) as well as sets for which the limit of ratios of consecutive elements diverges. In particular, this includes examples previously considered in work of Palac\'{i}n-Sklinos and Poizat. Finally, we show that stability of $(\mathbb{Z},+,0,A)$ implies a fair amount of sparsity for $A$. We use a result of Erd\H{o}s, Nathanson, and S\'{a}rk\"{o}zy to show that if $(\mathbb{Z},+,0,A)$ does not define the ordering on $\mathbb{Z}$, then the lower asymptotic density of any finitary sumset of $A$ is zero. We then show that stability of $(\mathbb{Z},+,0,A)$ implies certain restrictions on the behavior of finite arithmetic progressions in finitary sumsets of $A$.