Lacunary Arithmetic convergence

TL;DR

This paper introduces the lacunary arithmetic convergent sequence space $AC_{\theta}$, explores its relations with other sequence spaces, and extends the concept using modulus functions to derive new properties.

Contribution

It defines the new sequence space $AC_{\theta}$, investigates its relationship with $AC_{\sigma_1}$, and introduces $AC_{\theta}(f)$ using modulus functions, expanding the theory of arithmetic convergence.

Findings

Defined the sequence space $AC_{\theta}$.

Established relations between $AC_{\theta}$ and $AC_{\sigma_1}$.

Extended the concept to $AC_{\theta}(f)$ with modulus functions.

Abstract

In this article we introduce and study the lacunary arithmetic convergent sequence space $AC_{\theta}$. Using the idea of strong Ces\`{a}ro summable sequence and arithmetic convergence we define $AC_{\sigma_1}$ and study the relations between $AC_{\theta}$ and $AC_{\sigma_1}$. Finally using modulus function we define $AC_{\theta}(f)$ and study some interesting results.