Title

Sums in the grid

Abstract

Let A and B be down-sets in the grid [k]n = {0, ... , k - 1}n. Given the sizes of A and B, how small can A + B = {a + b ∈ [k]n: a ∈ A, b ∈ B} be? Our main aim in this paper is to give a best-possible lower bound for |A + B| in terms of |A| and |B|. For example, although if |A| = |B| = kn-1 we may have |A + B| = kn-1, we show that if |A| = |B| = kn-1 + 1 then |A + B| ≥ 2kn-1 + 1.

Publication Title

Discrete Mathematics

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