CEU Electronic Theses and Dissertations, 2011
| Author | Glasscock, Daniel Garrett |
|---|---|
| Title | Sumset Estimates in Abelian Groups |
| Summary | Let A be a finite set in an abelian group. We define A+A = {a_1+a_2 | a_1, a_2 in A }, A-A = {a_1-a_2 | a_1, a_2 in A } to be the sum set and difference set of A, respectively. The cardinalities of these sets are denoted by |A|, |A+A|, and |A-A|. Let |A|=n. It is immediate that n <= |A + A|, |A-A| <= n^2; it is not difficult to show that n <= |A+A| <= n(n+1)/2, n <= |A-A| <= n(n-1)+1 and that these bounds are achieved. If n, s, and d are positive integers such that n <= s <= n(n+1)/2 and n <= d <= n(n-1) + 1, then it is an easy exercise to construct sets A, A' in some abelian group with |A|=|A'|=n and |A+A| = s, |A'-A'| = d. It is in general, however, impossible to construct a single set A with |A|=n, |A+A| = s, |A-A|=d. In this thesis, we are concerned with understanding what triples (n,s,d) are attainable and generalizations of this problem. We explore the interplay between |A+A|, |A-A|, and the cardinalities of higher sumsets, as well as the cardinalities of sumsets involving multiple distinct sets. |
| Supervisor | Harcos, Gergely |
| Department | Mathematics MSc |
| Full text | https://www.etd.ceu.edu/2011/glasscock_daniel-garrett.pdf |
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