New York Number Theory Seminar
November 30, 2000, 5PM
Renling Jin, College of Charleston
- Title:
- Scrutinizing the Sumset Phenomenon
- Subtitle:
- The impact of the study of elusive nonstandard integers on the
study of finite integers in our real world.
- Abstract:
- The sumset phenomenon states that if both A and B are
large in terms of "measure", then A+B is not small in terms of
"order-topology". A well-known example of the phenomenon in real
analysis says that if both A and B have positive Lebesgue measure,
then A+B contains a non-empty open interval. Note that there exists
a nowhere dense set of positive Lebesgue measure. In the talk,
I will present two new examples of the sumset phenomenon about the
sequences of natural numbers and the sequences of finite subsets of
natural numbers. In these cases, I use certain kind of density for
"measure" and certain kind of syndeticity for "order-topology".
I will also introduce some of my more recent results which indicate
that the two examples mentioned above are optimal in some sense.
In the talk, I will also explain how we can prove these results
using nonstandard analysis in a natural way.