Some new results on higher energies

Shkredov, I.

doi:10.1090/S0077-1554-2014-00212-0

Some new results on higher energies
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by I. D. Shkredov
Translated by: Christopher Hollings

Trans. Moscow Math. Soc. 2013, 31-63

DOI: https://doi.org/10.1090/S0077-1554-2014-00212-0

Published electronically: April 9, 2014

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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 $|AA|$ and $|A(A+1)|$. New structural results, involving the notion of a dual popular difference set, are proved in terms of higher energies.

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