On Sums of Dilates

Let k be a positive integer and let A ⊂ Z. We let k · A = {ka : a ∈ A} denote the k-dilation of A, and let kA = A+ · · ·+ A (k-times) be the k-fold sumset of A. We observe that A+ k · A ⊂ A+ kA = (k + 1)A and that, in general, A+ k · A is much smaller than (k + 1)A. It is well known that |(k + 1)A| (k + 1)|A| − k, and that equality holds only if A is an arithmetic progression. Indeed, if A is an arithmetic progression with |A| k, one can check that A+ k · A = (k + 1)A. So it is a natural problem to study lower bounds for |A+ k · A| as well as the description of the extremal cases. The case k = 1 is trivial since |A+ A| 2|A| − 1 and equality holds for arithmetic progressions. The case k = 2 (see [3]) is also easy since we can split A = A1 ∪ A2 into the two classes (mod 2), and then |A+ 2 · A|=|A1 + 2 · A|+ |A2 + 2 · A| |A1|+ |2 · A| − 1 + |A2|+ |2 · A| − 1 = 3|A| − 2. (If A contains only one class we write A = 2 · A′ + i and then