Characterization of ideals equating ideal-limit and ideal-cluster Tauberian properties

Characterize, for a given positive rate sequence r tending to zero with divergent sum, the class of ideals on the nonnegative integers for which the conditions that every real sequence with successive differences O(r_n) satisfies \Lambda_x(\mathcal{I})=\mathrm{L}_x and that every such sequence satisfies \Gamma_x(\mathcal{I})=\mathrm{L}_x are equivalent.

Background

For an ideal \mathcal{I} on the nonnegative integers and a real sequence x, \mathrm{L}_x denotes the set of ordinary accumulation points, \Gamma_x(\mathcal{I}) the set of \mathcal{I}-cluster points, and \Lambda_x(\mathcal{I}) the set of \mathcal{I}-limit points. Theorem 2.1 establishes that, for a positive sequence r tending to zero with divergent sum, the property \Lambda_x(\mathcal{I})=\mathrm{L}_x for every sequence satisfying |\Delta x_n|=O(r_n) implies the corresponding cluster-point property \Gamma_x(\mathcal{I})=\mathrm{L}_x, and that the latter is equivalent to the equivalence between \mathcal{I}-convergence and ordinary convergence under the same Tauberian bound.

The theorem further proves equivalence of the limit-point and cluster-point properties when \mathcal{I} is a P^+-ideal. The paper notes that the equivalence can fail for the ideal of asymptotic density-zero sets \mathcal{Z}, so a general characterization for arbitrary ideals remains unresolved.