Iterated sumsets and Hilbert functions

Shalom Eliahou (1) , Eshita Mazumdar

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

Let A be a finite subset of an abelian group (G, +). Let h ≥ 2 be an integer. If |A| ≥ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + · · · + A is known, what can one say about |(h − 1)A| and |(h + 1)A|? It is known that |(h − 1)A| ≥ |hA| (h−1)/h , a consequence of Plünnecke's inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h≥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| ≥ θ(x, h) |hA| (h−1)/h for some factor θ(x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that θ(x, h) asymptotically tends to e ≈ 2.718 as |A| grows and h lies in a suitable range varying with |A|.

Mots clés: Plünnecke's inequality; Standard Graded Algebra; Macaulay's Theorem; Stirling's formula

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]; Algèbre commutative [math.AC]

Shalom Eliahou, Eshita Mazumdar. Iterated sumsets and Hilbert functions. 2020. ⟨hal-02867527v1⟩