It is known that a set A ⊂ ℕ belongs to the density ideal \(\mathcal{Z}\) if and only if zero is a dispersion point of a specific interval subset of ℝ defined for the sequence 1/n, namely, of the set \({\bigcup }_{i\in A}\left[1/\left(i+1\right),1/i\right].\) We investigate here for which decreasing sequences (xn) tending to zero the analogous equivalence holds, that is, for which sequences A ∈ \(\mathcal{Z}\) if and only if zero is a dispersion point of \({\bigcup }_{i\in A}\left[{x}_{i+1},{x}_{i}\right].\)
