Sumsets of sequences in abelian groups and flags in field extensions

TL;DR

This paper investigates the minimal functions associated with sumsets in finite abelian groups and flags in field extensions, classifying their structure for certain group sizes and constructing examples for others.

Contribution

It provides a classification of minimal sumset functions for specific group sizes and introduces a polyhedral framework to analyze these functions.

Findings

Classified minimal functions for groups of size less than 18, prime powers, and products of two primes.

Constructed explicit orderings with non-classifiable minimal functions for other sizes.

Developed a polyhedral model encoding the data of the sumset functions.

Abstract

For a finite abelian group $G$ with subsets $A$ and $B$, the sumset $AB$ is $\{ab \mid a\in A, b \in B\}$. A fundamental problem in additive combinatorics is to find a lower bound for the cardinality of $AB$ in terms of the cardinalities of $A$ and $B$. This article addresses the analogous problem for sequences in abelian groups and flags in field extensions. For a positive integer $n$, let $[n]$ denote the set $\{0,\dots,n-1\}$. To a finite abelian group $G$ of cardinality $n$ and an ordering $G = \{1=v_0,\dots,v_{n-1}\}$, associate the function $T \colon [n] \times [n] \rightarrow [n]$ defined by \[ T(i,j) = \min\big\{k \in [n] \mid \{v_0,\dots,v_i\}\{v_0,\dots,v_j\} \subseteq \{v_0,\dots,v_k\}\big\}. \] Under the natural partial ordering, what functions $T$ are minimal as $\{1=v_0,\dots,v_{n-1}\}$ ranges across orderings of finite abelian groups of cardinality $n$? We also ask the analogous question for degree $n$ field extensions. We explicitly classify all minimal $T$ when $n < 18$, $n$ is a prime power, or $n$ is a product of $2$ distinct primes. When $n$ is not as above, we explicitly construct orderings of abelian groups whose associated function $T$ is not contained in the above classification. We also associate to orderings a polyhedron encoding the data of $T$.