Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions

TL;DR

This paper investigates properties of simple modular density ideals generated by weight and modulus functions, revealing the number of functions producing the same ideal and exploring their structural characteristics.

Contribution

It demonstrates the uniqueness or multiplicity of functions generating the same ideal and explores the interaction between submeasures, functions, and ideals in this context.

Findings

For unbounded modulus functions, there are either one or continuum many functions generating the same ideal.

Established relationships between submeasures and generating functions of the ideals.

Provided observations on increasing-invariance properties of the ideals.

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, $g\in G$, $G=\{g:\omega \to [0,\infty):\frac{k}{g(k)}\not\to 0 \text{ and }\:\: g(k)\to \infty \text{ as }\:\:k\to \infty \}$ 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 $\{\phi_k\}_{k\in \omega}$ 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.