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Roth’s theorem in many variables

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Abstract

We prove that if A ⊆ {1, ..., N} has no nontrivial solution to the equation x 1 + x 2 + x 3 + x 4 + x 5 = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \), c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614, Poznań, Poland

    Tomasz Schoen

  2. Division of Algebra and Number Theory, Steklov Mathematical Institute, ul. Gubkina, 8, Moscow, Russia, 119991

    Ilya D. Shkredov

  3. Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Sovetskaya str. 14, Yaroslavl, Russia, 150000

    Ilya D. Shkredov

  4. IITP RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994

    Ilya D. Shkredov

Authors
  1. Tomasz Schoen
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  2. Ilya D. Shkredov
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Corresponding author

Correspondence to Tomasz Schoen.

Additional information

The author is partly supported by MNSW grant N N201 543538.

The author is supported by grant RFFI NN 11-01-00759, Russian Government project 11.G34.31.0053, Federal Program “Scientific and scientific-pedagogical staff of innovative Russia” 2009–2013, grant mol a ved 12-01-33080 and grant Leading Scientific Schools N 2519.02012.1.

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Schoen, T., Shkredov, I.D. Roth’s theorem in many variables. Isr. J. Math. 199, 287–308 (2014). https://doi.org/10.1007/s11856-013-0049-0

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  • DOI: https://doi.org/10.1007/s11856-013-0049-0

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