Towards a function field version of Freiman's Theorem

TL;DR

This paper explores a function field analogue of Freiman's Theorem, characterizing subspaces with small product spaces and linking them to genus 0 or 1 function fields, advancing understanding of additive structures in algebraic function fields.

Contribution

It provides a complete characterization of subspaces with doubling dimension twice that of the original space, connecting them to genus 0 or 1 function fields and Riemann-Roch spaces.

Findings

Spaces with   dimension are included in genus 0 or 1 function fields.

Such spaces are characterized up to multiplication by a constant field element.

If the genus is 1, the space is a Riemann-Roch space.

Abstract

We discuss a multiplicative counterpart of Freiman's $3k-4$ theorem in the context of a function field $F$ over an algebraically closed field $K$. Such a theorem would give a precise description of subspaces $S$, such that the space $S^2$ spanned by products of elements of $S$ satisfies $\dim S^2 \leq 3 \dim S-4$. We make a step in this direction by giving a complete characterisation of spaces $S$ such that $\dim S^2 = 2 \dim S$. We show that, up to multiplication by a constant field element, such a space $S$ is included in a function field of genus $0$ or $1$. In particular if the genus is $1$ then this space is a Riemann-Roch space.