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Abstract: We introduce a new probabilistic technique for finding -almost-periods- ofconvolutions of subsets of groups. This gives results similar to theBogolyubov-type estimates established by Fourier analysis on abelian groups butwithout the need for a nice Fourier transform to exist. We also presentapplications, some of which are new even in the abelian setting. These includea probabilistic proof of Roth-s theorem on three-term arithmetic progressionsand a proof of a variant of the Bourgain-Green theorem on the existence of longarithmetic progressions in sumsets A+B that works with sparser subsets of {1,

., N} than previously possible. In the non-abelian setting we exhibitanalogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additivecombinatorics, showing that product sets A B C and A^2 A^{-2} are ratherstructured, in the sense that they contain very large iterated product sets.This is particularly so when the sets in question satisfy small-doublingconditions or high multiplicative energy conditions. We also present results onstructures in product sets A B. Our results are -local- in nature, meaning thatit is not necessary for the sets under consideration to be dense in the ambientgroup. In particular, our results apply to finite subsets of infinite groupsprovided they -interact nicely- with some other set.



Author: Ernie Croot, Olof Sisask

Source: https://arxiv.org/







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