Iterated sumsets and Hilbert functions

Let A be a finite subset of an abelian group (G, +). For h is an element of N, let hA = A + ... + A denote the h-fold iterated sumset of A. If vertical bar A vertical bar >= 2, understanding the behavior of the sequence of cardinalities |hA| is a fundamental problem in additive combinatorics. For instance, if vertical bar hA vertical bar is known, what can one say about vertical bar(h - 1)A vertical bar and vertical bar(h + 1)A vertical bar? The current classical answer is given by vertical bar(h- 1)A vertical bar >= vertical bar hA vertical bar((h-1)/h), a consequence of Plunnecke's inequality based on graph theory. We tackle here this problem with a completely new approach, namely by invoking Macaulay's classical 1927 theorem on the growth of Hilbert functions of standard graded algebras. With it, we first obtain demonstrably strong bounds on |hA| as h grows. Then, using a recent condensed version of Macaulay's theorem, we derive the above Plunnecke-based estimate and significantly improve it in the form vertical bar(h- 1)A vertical bar >= theta(x, h) vertical bar hA vertical bar((h-1)/h) for h >= 2 and some explicit factor theta(x, h) > 1, where x is an element of R satisfies x >= h and vertical bar hA vertical bar = (x/h). Equivalently and more simply,vertical bar(h- 1)A vertical bar >= h/x vertical bar hA vertical bar.We show that theta(x, h) often exceeds 1.5 and even 2, and asymptotically tends to e 2.718 as x grows and h lies in a suitable range depending on x. (C)2021 Elsevier Inc. All rights reserved.