Roth’s theorem in ℤ4n

Sanders, Tom

doi:10.2140/apde.2009.2.211

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Roth's theorem in $\mathbb{Z}_4^n$

Tom Sanders

Vol. 2 (2009), No. 2, 211–234

DOI: 10.2140/apde.2009.2.211

Abstract

We show that if $A \subset ℤ_{4}^{n}$ contains no three-term arithmetic progressions in which all the elements are distinct then $| A | = o (4^{n} ∕ n)$ .

Keywords

Roth–Meshulam, cap set problem, Fourier, Freĭman, Balog–Szemerédi, characteristic 2, $\mathbb Z_4^n$, three-term arithmetic progressions

Mathematical Subject Classification 2000

Primary: 42A05

Milestones

Received: 14 November 2008

Revised: 30 March 2009

Accepted: 4 May 2009

Published: 1 May 2009

Authors

Tom Sanders
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge, CB3 0WA United Kingdom
http://www.dpmms.cam.ac.uk/~tws22/

Vol. 2, No. 2, 2009