Erd\H{o}s-Ulam ideals vs. simple density ideals

Abstract

The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function $g\colon\omega\to [0,\infty)$ such that $\lim_{n\to\infty}g(n)=\infty$ and $\frac{n}{g(n)}$ does not converge to $0$. Then the family $\mathcal{Z}_g=\{A\subseteq\omega:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal called simple density ideal (or ideal associated to upper density of weight $g$). We compare this class of ideals with Erd\H{o}s-Ulam ideals. In particular, we show that there are $\sqsubseteq$-antichains of size $\mathfrak{c}$ among Erd\H{o}s-Ulam ideals which are and are not simple density ideals. We characterize simple density ideals which are Erd\H{o}s-Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erd\H{o}s-Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal $\mathcal{I}$, asserts that given $B\in\mathcal{I}$ and $C\subseteq\omega$ with $\text{card}(C\cap n)\leq\text{card}(B\cap n)$ for all $n$, we have $C\in\mathcal{I}$. Finally, we pose some open problems.