Iterated sumsets and Hilbert functions
Résumé
Let $A$ be a finite subset of an abelian group $(G,+)$. For $h \in \mathbb{N}$, let $hA=A+\dots+A$ denote the $h$-fold iterated sumset of $A$. If $|A| \ge 2$, understanding the behavior of the sequence of cardinalities $|hA|$ is a fundamental problem in additive combinatorics. For instance, if $|hA|$ is known, what can one say about $|(h-1)A|$ and $|(h+1)A|$? The current classical answer is given by $$|(h-1)A| \ge |hA|^{(h-1)/h},$$ a consequence of Plünnecke'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 Plünnecke-based estimate and significantly improve it in the form
$$|(h-1)A| \ge \theta(x,h)\hspace{0.4mm}|hA|^{(h-1)/h}$$ for $h \ge 2$ and some explicit factor $\theta(x,h) > 1$, where $x \in \mathbb{R}$ satisfies $x \ge h$ and $|hA|=\binom{x}{h}$. Equivalently and more simply,
$$
|(h-1)A| \ge \frac hx\: |hA|.
$$
We show that $\theta(x,h)$ often exceeds $1.5$ and even $2$, and asymptotically tends to $e\approx 2.718$ as $x$ grows and $h$ lies in a suitable range depending on $x$.
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