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This article is cited in 11 scientific papers (total in 11 papers)
Roth's theorem in many variables
T. Schoena, I. D. Shkredovbcd
a Faculty of Mathematics and Computer Science,
Adam Mickiewicz University, Umultowska 87, 61-614 Poznán, Poland
b Division of Algebra and Number Theory, Steklov Mathematical Institute,
ul. Gubkina, 8, Moscow, Russia, 119991
c IITP RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994
d Delone Laboratory of Discrete and Computational Geometry,
Yaroslavl State University, Sovetskaya str. 14, Yaroslavl, Russia, 150000
Abstract:
We prove that if $A\subseteq\{1,\dots,N\}$ 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.
Received: 25.10.2011
Revised: 30.10.2012
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