D* Sets and AIP* Sets in Z and Countable Fields

We consider the set of ultrafilter in Z, denoted beta Z. An IP set in Z is a set that contains some infinite sequence and all of its finite sums. An IP* set is a set that meets every IP set non-trivially. An AIP* set is a set A having he property that for some set B of zero upper Banach density, A cup B is IP*. Alternately, call a set IP rich if it is still IP upon the removal of any zero upper Banach density set. Then a set is AIP* if and only if it intersects every IP rich set non-trivially. Some ultrafilters have the property that all of their members have positive upper Banach density. Such ultrafilters are called essential and their members are called D sets. A set that ia a member of every essential idempotent ultrafilter is called a D* set. V. Bergelson asked whether or not every D* set is AIP*. Equivalently, whether every D set is IP rich. We give a negative answer to this question. That is, we construct a set A that is an IP rich set but not a D set. We also extend our construction to countably infinite fields of positive characteristic