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LINEAR FORMS AND QUADRATIC UNIFORMITY FOR FUNCTIONS ON 𝔽np

Published online by Cambridge University Press:  07 March 2011

W. T. Gowers
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, U.K. (email: w.t.gowers@dpmms.cam.ac.uk)
J. Wolf
Affiliation:
Department of Mathematics, Rutgers The State University of New Jersey, 110 Frelinghuysen Rd., Piscataway, NJ 08854, U.S.A. (email: julia.wolf@cantab.net)
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Abstract

We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176], which shows that a system of linear forms on 𝔽np with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of 𝔽np. While in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153], we use the Hahn–Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the U3 inverse theorem [B. J. Green and T. Tao, An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153].

Type
Research Article
Copyright
Copyright © University College London 2011

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