Iterated sumsets and Hilbert functions

TL;DR

This paper improves bounds on the sizes of iterated sumsets in abelian groups by modeling their growth with Hilbert functions and applying Macaulay's theorem, revealing asymptotic behavior as the set size increases.

Contribution

It introduces a novel approach using Hilbert functions and Macaulay's theorem to refine bounds on sumset sizes, surpassing previous inequalities.

Findings

New bound with factor θ(x,h) > 1 for sumset sizes

Asymptotic behavior of θ(x,h) approaches e as |A| grows

Enhanced understanding of sumset growth in abelian groups

Abstract

Let A be a finite subset of an abelian group (G, +). Let h $\ge$ 2 be an integer. If |A| $\ge$ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + $\times$ $\times$ $\times$ + A is known, what can one say about |(h -- 1)A| and |(h + 1)A|? It is known that |(h -- 1)A| $\ge$ |hA| (h--1)/h , a consequence of Pl{\"u}nnecke's inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h$\ge$0 with the Hilbert function of a standard graded algebra. We then apply Macaulay's 1927 theorem on the growth of Hilbert functions, and more specifically a recent condensed version of it. Our bound implies |(h -- 1)A| $\ge$ $\theta$(x, h) |hA| (h--1)/h for some factor $\theta$(x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that $\theta$(x, h) asymptotically tends to e $\approx$ 2.718 as |A| grows and h lies in a suitable range varying with |A|.