Iterated-sumset spectra: The complete exponent law and its rank geometry

Abstract

For integers $h,k\geq 1$, let $hA$ be the $h$-fold sumset of $A$ and put $\mathcal{R}(h,k)=\{|hA|:A\subset\mathbb{Z}, |A|=k\}$. Previously, the fixed-cardinality exponent law was known only for $k\leq 3$; every fixed $k\geq 4$ remained open. We settle the problem in full by determining the complete fixed-cardinality exponent law: $|\mathcal{R}(h,k)|=\begin{cases}1,&k\leq 2,\\ h,&k=3,\\ h^{k-1+o_k(1)},&k\geq 4\end{cases}$. Here $o_k(1)\to 0$ as $h\to\infty$ with $k$ fixed. More sharply, for fixed $k\geq 4$, an interval of length $Θ_k(h^{k-1})$ contains at least $h^{k-1-o_k(1)}$ attainable values. At $k=4$ we prove $|\mathcal{R}(h,4)|=Θ(h^3)$ with positive lower density in its ambient interval, disproving Nathanson's proposed $o(h^3)$ and $O(h^2)$ bounds. One bounded addition-table geometry drives these results, coupling Hilbert-energy amplification to optimal finite-observation compression. Every ordered real $k$-set ($k\geq 2$) has an integer model in $[0,O_k(h^{k-2})]$ preserving every sum equality and strict comparison through degree $h$; the exponent $k-2$ is sharp. The universal label-realization length is therefore $Θ_k(h^{k-2})$, one power sharper than Nathanson's $O_k(h^{k-1})$ bound. For $h\geq 2$ and $k\geq 3$, minimum active rank equals realization-frequency codimension, exponent-shape codimension, and sampling-rarity exponent; a full-exponent family has maximal-rank witnesses with Cohen-Macaulay toric coordinate rings. At rank zero, for $h\geq 2$, it proves the conjectural OEIS A227589 formula $\binom{h+2}{2}+\mathbf{1}_{\{2\nmid h\}}$ for the least normalized diameter of a four-point $B_h$-set. It also gives exact fixed-$(h,k)$ popularity laws for $k$-subsets of $\{1,\ldots,q\}$ as $q\to\infty$, resolving Nathanson's Problems 9 and 10.