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On growth in an abstract plane


Authors: Nick Gill, Harald A. Helfgott and Misha Rudnev
Journal: Proc. Amer. Math. Soc. 143 (2015), 3593-3602
MSC (2010): Primary 51A35; Secondary 05B25
Published electronically: April 13, 2015
MathSciNet review: 3348800
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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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Additional Information

Nick Gill
Affiliation: Department of Mathematics, University of South Wales, Treforest, CF37 1DL, United Kingdom
Email: nicholas.gill@southwales.ac.uk

Harald A. Helfgott
Affiliation: Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, F-75230 Paris, France
Email: helfgott@dma.ens.fr

Misha Rudnev
Affiliation: School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom
Email: m.rudnev@bristol.ac.uk

DOI: https://doi.org/10.1090/proc/12309
Received by editor(s): December 21, 2012
Published electronically: April 13, 2015
Additional Notes: The first author would like to thank the University of Bristol, to which he has been a frequent visitor during the writing of this paper.
The second author thanks MSRI (Berkeley) for its support during a stay there.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.