Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Discrete Mathematics
| Hi-index | 0.00 |
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.