Relating S_{I,ε} to Gajda–Miller families N_J^* and N_J

Clarify the relationships—containments or equalities—between the layered σ-ideals S_{I,ε} and the families N_J^* and N_J (defined from an ideal J on ω) introduced by Gajda and Miller, for varying choices of partition I of ω into finite nonempty intervals and ε ∈ ℓ^1_+.

Background

The concluding remarks note prior work of Gajda and Miller that defines intermediate families N_J^* and N_J with E ⊆ N_J^* ⊆ N_J ⊆ N. Given the placement of S_{I,ε} between E and N in the paper’s inclusion diagram, it is natural to ask how S_{I,ε} compares to N_J^* and N_J.

Answering this would situate the layered ideals within the broader landscape of definable smallness notions and could reveal new structural or combinatorial characterizations.