For a finite set of integers A of size k and an integer h at least 2, the h-fold sumset hA collects every sum of h elements of A with repetitions allowed, and its size is at least hk - h + 1, with equality exactly when A is an arithmetic progression. This Idea covers the next layers of that inverse problem: what A must look like when |hA| exceeds the minimum by only a bounded amount. The answer is that A, after normalising by translation and by the gcd of its differences, is an interval of integers with one or two elements deleted, and which elements are deleted determines |hA| exactly. Learners see how a lower bound on |hA| bounds the diameter of A, how deleting an interior element of an interval leaves the sumset a full interval, and how a gap in the attainable values of |hA| rules out certain sizes altogether.
For a finite set of integers A = {a_0, ..., a_{k-1}} and an integer h >= 2, the h-fold sumset hA is the set of all sums of h elements of A with repetitions allowed. Its normal form A(N) is obtained b…