Sums in the grid

@article{Bollobs1996SumsIT,
  title={Sums in the grid},
  author={B{\'e}la Bollob{\'a}s and Imre Leader},
  journal={Discrete Mathematics},
  year={1996},
  volume={162},
  pages={31-48}
}
Let A and B be down-sets in the grid [k]" = {0 . . . . . k 1}". Given the sizes of A and B, how small can A + B = {a + be[k]": aeA , beB} be? Our main aim in this paper is to give a best-possible lower bound for I A + B I in terms of I A I and I B I. For example, although if [A I = I B I = k"x we may have I Z + B I = k"1, we show that if IAI = [BI = k "-1 + 1 then IA + BI >/2k "-1 + 1. 
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