Sets with arithmetic progressions are abundant

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.