Energies and Structure of Additive Sets

Shkredov Ilya


In this paper we prove that any sumset or difference set has large $\textsf{E}_3$ energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as $\textsf{E}_k$, $\textsf{T}_k$ and Gowers norms. In particular, we give criteria for a set to be a 

  • set of the form $H\dotplus \Lambda$, where $H+H$ is small and $\Lambda$ has "random structure",
  • set equal to a disjoint union of sets $H_j$ each with small doubling,
  • set having a large subset $A'$ with $2A'$ equal to a set with small doubling and $|A'+A'| \approx |A|^4 / \textsf{E}(A)$.


Additive combinatorics, sumsets, energies

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