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Erd\H{o}s-Ulam ideals vs. simple density ideals. (arXiv:1711.03578v2 [math.CO] UPDATED)
来源于:arXiv
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 查看全文>>