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Some new results on higher energies


Author: I. D. Shkredov
Translated by: Christopher Hollings
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 74 (2013), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2013, 31-63
MSC (2010): Primary 11B30; Secondary 11B75
DOI: https://doi.org/10.1090/S0077-1554-2014-00212-0
Published electronically: April 9, 2014
MathSciNet review: 3235789
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Abstract: This article is concerned with the method of higher energies from combinatorial number theory. Upper bounds are obtained for the additive energies of convex sets and of sets $ A$ with small $ \vert AA\vert$ and $ \vert A(A+1)\vert$. New structural results, involving the notion of a dual popular difference set, are proved in terms of higher energies.


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Additional Information

I. D. Shkredov
Affiliation: Department of Algebra and Number Theory, Russian Academy of Sciences, Steklov Mathematical Institute, Moscow; Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Yaroslavl; Kharkevich Institute of Information Transmission Problems, Russian Academy of Sciences
Email: ilya.shkredov@gmail.com

DOI: https://doi.org/10.1090/S0077-1554-2014-00212-0
Keywords: Combinatorial number theory, higher energies, popular difference set
Published electronically: April 9, 2014
Additional Notes: This work was supported by the grant RFFI 11-01-00759, the Government grant RF 11.G34.31.0053, the Federal Program “Scientific and Scientific-Pedagogical Personnel in Russia” 2009–2013, the Grant for Scientific Projects Undertaken by Leading Youth Collectives 12-01-33080, and the Grant for Leading Scientific Schools 2519.2012.1.
Article copyright: © Copyright 2014 American Mathematical Society