71 research outputs found
On Heilbronn's exponential sum
In the paper we prove a new upper bound for Heilbronn's exponential sum and
obtain some applications of our result to distribution of Fermat quotients.Comment: 10 page
Energies and structure of additive sets
In the paper we prove that any sumset or difference set has large E_3 energy.
Also, we give a full description of families of sets having critical relations
between some kind of energies such as E_k, T_k and Gowers norms. In particular,
we give criteria for a set to be a 1) set of the form H+L, where H+H is small
and L has "random structure", 2) set equals a disjoint union of sets H_j, each
H_j has small doubling, 3) set having large subset A' with 2A' is equal to a
set with small doubling and |A'+A'| \approx |A|^4 / \E(A).Comment: 52 page
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