Sumsets

Sumsets are a central notion in additive combinatorics. For subsets A and B of an abelian group, their sumset A+B is the set {a+b : a in A, b in B}. In particular, A+B = B+A, and translating both sets by a fixed element t satisfies (A+t)+(B+t) = (A+B)+t. The k-fold sumset kA is A added to itself k times, and the difference set A−B = {a−b} is defined similarly.

For finite, nonempty A,B, there are basic size bounds: |A+B| ≥ |A|+|B|−1 when A,B ⊆ Z. Consequently, by induction, |kA| ≥ k|A|−(k−1) for all k≥1. In contrast, in general abelian groups different bounds may hold (e.g., Cauchy–Davenport in Z_p).

Example: A={0,1}, B={0,2} yields A+B={0,1,2,3}, so |A+B|=4 ≥ |A|+|B|−1=3. Sumsets can grow rapidly; small doubling often indicates additive structure.

Extensions and topics: sum-free sets are those with A ∩ (A+A) = ∅. Freiman's theorem describes the structure of finite sets with small doubling |A+B| relative to |A|. More generally, sumsets are studied for subsets of integers, Euclidean space, and arbitrary abelian groups, with variants such as multi-set sums, restricted sums, and weighted sums.