Freiman's $3k-4$ Theorem for Function Fields

TL;DR

This paper extends Freiman's $3k-4$ Theorem from integers to function fields, showing that small product sets imply the structure of the generating space is closely related to a small genus function field.

Contribution

It provides a novel function field analogue of Freiman's theorem, linking small product set dimension to the structure of function fields and Riemann-Roch spaces.

Findings

Small product set dimension implies the generated function field has small genus.

The generating space is of small codimension inside a Riemann-Roch space.

The result generalizes additive combinatorics to algebraic function fields.

Abstract

Freiman's $3k-4$ Theorem states that if a subset $A$ of $k$ integers has a Minkowski sum $A+A$ of size at most $3k-4$, then it must be contained in a short arithmetic progression. We prove a function field analogue that is also a generalisation: it states that if $K$ is a perfect field and if $S\supset K$ is a vector space of dimension $k$ inside an extension $F/K$ in which~$K$ is algebraically closed, and if the $K$-vector space generated by all products of pairs of elements of $S$ has dimension at most $3k-4$, then $K(S)$ is a function field of small genus, and $S$ is of small codimension inside a Riemann-Roch space of $K(S)$.