Lacunary arithmetic statistical convergence

A lacunary sequence is an increasing integer sequence $\theta =(k_r)$ such that $k_r-k_{r-1}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }\mathrm{.}$ In this article we introduce arithmetic statistically convergent sequence space $ASC$ and lacunary arithmetic statistically convergent sequence space $AS{C}_{\theta }$ and study some inclusion properties between the two spaces. Finally we introduce lacunary arithmetic statistical continuity and establish some interesting results.