# Sets with arithmetic progressions are abundant

## TL;DR

This paper proves that various large sets, including those with unsettled properties like J-sets and C-sets, are rich in arithmetic progressions, demonstrating their abundance of such progressions.

## Contribution

The paper provides an elementary proof that sets of A.P. rich contain arbitrarily long arithmetic progressions, extending known results to broader classes of sets.

## Key findings

- Sets of A.P. rich contain arbitrarily long arithmetic progressions.
- Elementary proof established for the abundance of progressions in these sets.
- Includes sets like J-sets, C-sets, D-sets, previously with unsettled properties.

## Abstract

Furstenberg, Glasscock, Bergelson, Beiglboeck have been studied abundance in arithmatic progression on various large sets like piecewise syndetic, central, thick, etc. but also there are so many sets in which abundance in progression is still unsettled like J-sets, C-sets, D-sets etc. But all of these sets have a common property that they contains arbitrary length of arithmatic progressions. These type of sets are called sets of A.P. rich, we have given an elementary proof of abundance of those sets.