Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions
Pratulananda Das, Subhankar Das
TL;DR
This work studies modular simple density ideals $Z_g(f)$ formed by combining weight functions $g$ in $G$ with unbounded modulus functions $f$. It shows that for a fixed $f$, the set of $g$ producing the same ideal is either a singleton or has cardinality $\mathfrak{c}$, and it gives criteria for when two such ideals coincide in terms of the relative growth of $f(g(n))$ and $f(h(n))$, including the key relation $Z_g(f)=Z_{\max\{g,h\}}(f)$. The authors express $Z_g(f)$ as a generalized density ideal generated by a sequence of submeasures, providing a block-structured description via intervals $[k_m,k_{m+1})$ and a corresponding family of submeasures $\phi_m$, and they establish equivalences tying the boundedness of $\sup_k \phi_k(\omega)$ to the boundedness of a related growth sequence. They further explore increasing-invariance properties and show that $Z_g(f)$ fits within the landscape of Erdős–Ulam ideals, demonstrating both inclusions and limitations through concrete constructions. Overall, the paper extends classical density- and weight-based ideals to a modular setting, clarifying when different weight functions yield the same combinatorial object and how these ideals interact with invariant and EU-structure notions.
Abstract
In this article our main object of investigation is the simple modular density ideals $\mathcal{Z}_g(f)$ introduced in [Bose et al., Indag. math., 2018] where $g$ is a weight function, more precisely, $g\in G$, $G=\{g:ω\to [0,\infty):\frac{k}{g(k)}\not\to 0 \text{ and }\:\: g(k)\to \infty \text{ as }\:\:k\to \infty \}$ and $f$ is an unbounded modulus function. We mainly investigate certain properties of these ideals in line of [Kwela et al, J. math. Anal. Appl., 2019]. For an unbounded modulus function $f$ it is shown that there are $1$ or $\ck$ many functions $g\in G$ generating the same ideal $\mathcal{Z}_g(f)$. We then obtain certain interactive results involving the sequence of submeasures $\{φ_k\}_{k\in ω}$ generating the ideal $\mathcal{Z}_g(f)$ and the functions $g,f$. Finally, we present some observations on $\mathcal{Z}_g(f)$ ideals related to the notion of increasing-invariance.