Sumsets and Veronese varieties

TL;DR

This paper links sumset growth in integer subsets to Veronese varieties, using algebraic geometry tools to determine polynomial formulas and bounds for sumset cardinalities and phase transitions.

Contribution

It introduces a novel association between sumsets and Veronese varieties, enabling geometric analysis of sumset growth and phase transition bounds.

Findings

Derived polynomial formulas for sumset cardinalities.

Established bounds for phase transition points.

Connected sumset properties with geometric features of Veronese varieties.

Abstract

In this paper, to any subset $\mathcal{A} \subset \mathbb{Z}^{n}$ we explicitly associate a unique monomial projection $Y_{n,d_{\mathcal{A}}}$ of a Veronese variety, whose Hilbert function coincides with the cardinality of the $t$-fold sumsets $t\mathcal{A}$. This link allows us to tackle the classical problem of determining the polynomial $p_{\mathcal{A}} \in \mathbb{Q}[t]$ such that $|t\mathcal{A}| = p_{\mathcal{A}}(t)$ for all $t \geq t_0$ and the minimum integer $n_0(\mathcal{A}) \leq t_0$ for which this condition is satisfied, i.e. the so-called {\em phase transition} of $|t\mathcal{A}|$. We use the Castelnuovo--Mumford regularity and the geometry of $Y_{n,d_{\mathcal{A}}}$ to describe the polynomial $p_{\mathcal{A}}(t)$ and to derive new bounds for $n_0(\mathcal{A})$ under some technical assumptions on the convex hull of $\mathcal{A}$; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties $Y_{n,d_{\mathcal{A}}}$.