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

中文翻译：

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Lacunary算术统计收敛

甲缺位序列是递增的整数序列\（\ THETA =（K_R）\） ，使得\（K_R-K_ {R-1} \ RIGHTARROW \ infty \）作为\（R \ RIGHTARROW \ infty。\）在本文中，我们介绍了算术统计收敛序列空间ASC和lacunary算术统计收敛序列空间\（ASC _ {\ theta} \），并研究了这两个空间之间的一些包含性质。最后，我们介绍了Lamunary算术统计连续性并建立了一些有趣的结果。