ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N.42–2019 (341–356) 341 On asymptotically lacunary statistical equivalent (Wijsman sense) set sequences via ideal and modulus function Wadei Al-Omeri Department of Mathematics Al-Balqa Applied University Salt, 19117 Jordan wadeimoon1@hotmail.com Abstract. For any non-trivial ideal I⊆ ℘(N), and any non-empty closed subset A k ⊆ X, where (X, ρ) is a metric space. Let f be a modulus function, The ob- jective of this paper is to introduce and study the new notation by using a mod- ulus function, I (W L ), I (fW ), I [WN ] L θ , I (Wf L ), I ( W θ f ), I (WS L ) and I (WS L θ ). Which are natural combinations of the definitions for asymptotically lacunary equiva- lent (W ijsman sense). In addition, some relations among these new notions are also obtained. 1. Introduction and preliminaries The idea of convergence of a real sequence had been extended to statistical convergence by Fast [7] (see, also [26]). as follows: if K is a subset of natural numbers N,K n will denote the set {κ ∈ K: κ  n} and |K n | will denote the cardinality of K n . Natural density of K [9] is given by δ(K) := lim n 1 n |K n | , if ¯ δ(K) = δ (K) then we say that the natural density of K exists and it is de- noted simply by δ(K). A sequence (x n ) of real numbers is said to be statistically convergent to L if for arbitrary ε> 0, the set K(ε)= n ∈ N : |x n − L|≥ ε has natural density zero. The concept of statistical convergence plays an important role in the summability theory and functional analysis. The relationship be- tween the summability theory and statistical convergence has been introduced by Schoenberg [26]. In [3], Borwein introduced and studied strongly summable functions. The concept of Wijsman statistical convergence is implementation of the concept of statistical convergence to sequences of sets presented by Nuray and Rhoades in 2012. Similar to this concept, the concept of Wijsman lacunary statistical convergence was presented by Ulusu and Nuray in 2012. Kostyrko et al. [11] generalized statistical convergence with the help of an admissible ideal I of subsets of N, the set of positive integers and called it I -convergence. Quite recently, Sava¸ s et al. [25] unified the notions of statistical convergence and I -convergence to introduce new concepts of I -statistical convergence.