John-type theorems for generalized arithmetic progressions and iterated sumsets

阿贝尔半群中迭代和集增长的最优界

Optimal Bounds on the Growth of Iterated Sumsets in Abelian Semigroups

本文为阿贝尔半群有限子集的迭代和集增长给出最优上界，证明源自Macaulay定理的界可达，构造基于多项式代数中的单项式理想及其通过Gröbner基的形变。

$\nmathbb{F}_p^n$ 中迭代和集扩张

Iterated sumset expansion in $\mathbb{F}_p^n$

本文证明在有限域上，若每个集合不包含于低次多项式零点集，则迭代和集可在O(p)项内扩张至全空间，并讨论随机集合及与EGZ问题的联系。

Iterated sumsets and Hilbert functions

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 + $\times$ $\times$ $\times$ + 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{ü}nnecke's inequality. Here we improve 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 c…

Sumset and inverse sumset theorems for Shannon entropy

Let $G = (G,+)$ be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated sumsets $kA$ and difference sets $A-B$, while the inverse sumset theory of Freiman, Ruzsa, and others characterises those finite sets $A$ for which $A+A$ is small. In this paper we establish analogous results in which the finite set $A \subset G$ is replaced by a discrete random variable $X$ taking values in $G$, and the cardinality $|A|$ is replaced by the Shannon entropy $\mathrm{Ent}(X)$. In parti…

迭代和集谱：完整指数律及其秩几何

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

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。