Iterated sumsets and Hilbert functions

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 $hA=A+\dots+A$ 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\"unnecke's inequality. we improved 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 $$|(h-1)A| \ge \theta(x,h)\hspace{0.4mm}|hA|^{(h-1)/h}$$ 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 $|A|$. This is a joint work with Prof. Shalom Eliahou.

commutative algebraalgebraic topologycombinatorics

Audience: researchers in the topic

Applications of Combinatorics in Algebra, Topology and Graph Theory

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