# Abundance of arithmetic progressions in some combiantorially large sets

## TL;DR

This paper investigates the presence and abundance of arithmetic progressions within certain large sets in combinatorics, extending known results to Quasi Central, J-sets, and C-sets.

## Contribution

It extends the understanding of arithmetic progression abundance to new classes of large sets like Quasi Central, J-sets, and C-sets, beyond previously studied sets.

## Key findings

- Established abundance results for Quasi Central sets.
- Extended arithmetic progression results to J-sets.
- Provided new insights into C-sets and their structure.

## Abstract

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k length arithmetic progression and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result was extended for arbitrary semigroups by Bergelson and Hindman, again using the structure of Stone-Cech compactification of general semigroup. However they provided the abundances for various types of large sets. But the abundances in in many large sets is still unknown. In this work we will provide the abundance in Quasi Central sets, J-sets and C-sets.