# Lacunary arithmetic statistical convergence

## TL;DR

This paper introduces new sequence spaces based on lacunary and arithmetic statistical convergence, studies their properties, and explores the concept of lacunary arithmetic statistical continuity.

## Contribution

It defines the spaces ASC and ASC_θ, investigates their inclusion relations, and introduces lacunary arithmetic statistical continuity with key results.

## Key findings

- Established inclusion properties between ASC and ASC_θ
- Defined lacunary arithmetic statistical continuity
- Proved several theorems on the properties of these spaces

## Abstract

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.