On growth in an abstract plane

Nick Gill, H. A. Helfgott, Misha Rudnev

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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.
Original languageEnglish
Pages (from-to)3593-3602
Number of pages10
JournalProceedings of the American Mathematical Society
Issue number8
Publication statusPublished - 13 Apr 2015


  • math.CO
  • 51E15 (primary), 51A35, 11B30 (secondary)


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