In this paper, we introduce the notions of asymptotical strong σ 2 σ2 -equivalence, -statistical equivalence, lacunary -equivalence and equivalence in Wijsman sense for double set sequences. Also, investigate some relations between these new notions.

Using the concept of Orlicz function and uncertainty theory, some new class lacunary convergent sequences defined by functions have been introduced with convergence concepts in this paper. Some topological properties sequence spaces along inclusion relations investigated.

Abstract Let $F \subseteq [0,1]$ be a set that supports probability measure $\mu $ with the property |\widehat{\mu }(t)| \ll (\log |t|)^{-A}$ for some constant A> 0 $. $\mathcal{A}= (q_n)_{n\in{\mathbb{N}}} sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A>2$, we establish quantitative inhomogeneous Khintchine-type theorem in which (1) points interest are restricted...