Sumsets of sequences in abelian groups and flags in field extensions

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 $T$.