Electronic Theses and Dissertations





Document Type


Degree Name

Doctor of Philosophy


Mathematical Sciences



Committee Member

Randall McCutcheon

Committee Member

Paul Balister

Committee Member

James Campbell


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.


Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.