AI 中文总结
本文解决了固定基数迭代和集谱的指数律问题，确定了完整规律，证明k=4时|ℛ(h,4)|=Θ(h^3)，还解决了Nathanson的相关问题，揭示了其秩几何性质。
AI 中文摘要
对于整数h,k≥1，设hA为集合A的h重和集，记ℛ(h,k)={|hA|:A⊂ℤ, |A|=k}。此前，固定基数的指数律仅在k≤3时已知，所有固定k≥4的情况均未解决。本文完全解决了该问题，确定了完整的固定基数指数律：当k≤2时，|ℛ(h,k)|=1；当k=3时，|ℛ(h,k)|=h；当k≥4时，|ℛ(h,k)|=h^{k-1+o_k(1)}，其中o_k(1)随h→∞（k固定）趋于0。更精确地，对于固定k≥4，长度为Θ_k(h^{k-1})的区间包含至少h^{k-1-o_k(1)}个可达值。在k=4时，本文证明|ℛ(h,4)|=Θ(h^3)，且在其环境区间内具有正下界密度，否定了Nathanson提出的o(h^3)和O(h^2)界。一项有界加法表几何驱动了这些结果，将希尔伯特能量放大与最优有限观测压缩相结合。每个有序实k元组（k≥2）都有一个在[0,O_k(h^{k-2})]内的整数模型，该模型在次数h内保留所有和相等关系与严格比较，指数k-2是精确的。因此，通用标签实现长度为Θ_k(h^{k-2})，比Nathanson的O_k(h^{k-1})界更精确一个幂次。对于h≥2且k≥3，最小有效秩等于实现频率余维数、指数形状余维数和采样稀有度指数；全指数族具有极大秩见证，带有Cohen-Macaulay toric坐标环。在秩零处，对于h≥2，本文证明了OEIS A227589的猜想公式：四点B_h集的最小归一化直径为binom(h+2,2)+1_{{2∤h}}。当q→∞时，本文还给出了{1,…,q}的k子集的精确固定(h,k)流行度规律，解决了Nathanson的第9和第10问题。
英文摘要
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.
Comments106 pages, 1 figure
