Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions

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.