  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.
