Approach to the density ideal by density of special testing sets on the real line

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 $${\bigcup }_{i\in A}\left[{x}_{i+1},{x}_{i}\right].$$