Name
Abstract | ||
|---|---|---|
We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B@?R^2 in terms of the minimum number h"1(A,B) of parallel lines covering each of A and B. We show that, if h"1(A,B)=s and |A|=|B|=2s^2-3s+2, then|A+B|=|A|+(3-2s)|B|-2s+1. More precise estimations are given under different assumptions on |A| and |B|. This extends the 2-dimensional case of the Freiman 2^d-Theorem to distinct sets A and B, and, in the symmetric case A=B, improves the best prior known bound for |A|=|B| (due to Stanchescu, and which was cubic in s) to an exact value. As part of the proof, we give general lower bounds for two-dimensional subsets that improve the two-dimensional case of estimates of Green and Tao and of Gardner and Gronchi, related to the Brunn-Minkowski Theorem. |
Year | DOI | Venue |
|---|---|---|
2010 | 10.1016/j.jcta.2009.06.001 | J. Comb. Theory, Ser. A |
Keywords | Field | DocType |
additive combinatorics,general lower bound,small sumset,two-dimensional set,two-dimensional subsets,multi-dimensional,brunn–minkowski,2-dimensional case,brunn-minkowski theorem,two-dimensional case,lower bound,sumsets,hyperplanes,nonempty subsets,minimum number h,different assumption,symmetric case,number theory,2 dimensional | Discrete mathematics,Multi dimensional,Combinatorics,Cardinality,Sumset,Parallel,Hyperplane,Mathematics | Journal |
Volume | Issue | ISSN |
117 | 2 | Journal of Combinatorial Theory, Series A |
Citations | PageRank | References |
0 | 0.34 | 2 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| David J. Grynkiewicz | 1 | 42 | 10.33 |
| Oriol Serra | 2 | 147 | 28.05 |