On growth in an abstract plane

TL;DR

This paper explores the concept of growth in an abstract projective plane, bridging ideas from arithmetic combinatorics and geometry, and proposes strategies for geometric proofs over finite fields.

Contribution

It introduces a framework for defining and proving growth in an abstract projective plane with minimal axioms, extending geometric growth concepts beyond classical settings.

Findings

Growth can be defined in an abstract projective plane.

Geometric proofs of growth are possible over finite fields.

Strategies for such proofs are discussed and developed.

Abstract

There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over $\mathbb{R}$ or $\mathbb{C}$, geometric statements on growth often have geometric proofs, what little is known over finite fields rests on arithmetic proofs. We discuss strategies for geometric proofs of growth over finite fields, and show that growth can be defined and proven in an abstract projective plane -- even one with weak axioms.