RUI: Inverse Problems for Finite and Infinite Sets, and Nonstandard Methods

College Of Charleston, Charleston SC

Investigators

Abstract

Inverse problems study the structure of a set A of natural numbers when the size of A plus A (the sum of two copies of A) is relatively small. During the late 1950's and early 1960's G. A. Freiman derived a series of theorems, which indicated that for a finite set A, if the size of A plus A is small, then A must have some arithmetic structure. In the proposal the principal investigator proposes to study the inverse phenomenon for generalizing Freiman's theorems for finite sets and discovering new theorems for infinite sets using nonstandard methods. One type of Freiman's results optimally characterizes the structure of A when the size of A plus A is less than 3 times the size of A minus 2. Since then many efforts have been made by various people to generalize Freiman's theorems with this type of condition. However, no optimal structural characterization of A had been obtained for the size of A plus A greater then 3 times the size of A minus 2, until recently. It seemed unexpected that nonstandard methods were brought in and made a break-through. With the help of nonstandard analysis the principal investigator was able to give an optimal characterization of the structure of A when the size of A plus A is upper bounded by c times the size of A for a constant c slightly greater than 3. The principal investigator believes that the potential power of the methods hasn't been fully uncovered. He proposes to further his investigation and derive the same kind of characterization for the structure of A when the size of A plus A is less then the ten-thirds the size of A or further. The principal investigator also proposes to work on other problems including inverse problems for infinite sets using the nonstandard methods. Freiman's inverse phenomenon revealed a fundamental behavior of natural numbers. This idea had many applications in various fields such as combinatorial number theory, algebra, coding theory, integer programming, probability, etc. as mentioned in "Structure Theory of Set Addition", Asterisque No. 258 (1999), Societe Mathematique de France, Paris. Any improvement on Freiman's theorems would certainly bring about better applications in these fields. Since numbering and counting are fundamental to mathematics as well as our daily lives, a better understanding of our number system would obviously be beneficial to the both. One distinctive feature of this proposal is the use of the nonstandard methods. Nonstandard analysis uses the techniques in logic to create a world in which infinitely large numbers are allowed. An interesting aspect of this research is that these infinitely large numbers, which are purely imaginary to the human minds, could be used to prove theorems in the real world. This interaction between logic and number theory demonstrates the importance of interdisciplinary research. This research should also be instructive to young researchers encouraging them to open their minds, embrace different kinds of knowledge, and uncover the hidden relationships between these different disciplines.