On growth in an abstract plane

Gill, Nick; Helfgott, Harald; Rudnev, Misha

doi:10.1090/proc/12309

On growth in an abstract plane
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by Nick Gill, Harald A. Helfgott and Misha Rudnev PDF

Proc. Amer. Math. Soc. 143 (2015), 3593-3602 Request permission

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.

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