A Probabilistic Technique for Finding Almost-Periods of Convolutions

By Ernie Croot, Olof Sisask. 2010

122 citations recorded by the source corpus.

We introduce a new probabilistic technique for finding ‘almost-periods’ of convolutions of subsets of groups. This gives results similar to the Bogolyubovtype estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth’s theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain–Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, . . . , N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov–Freiman–Halberstam–Ruzsa-type results of additive combinatorics, showing that product sets A1 · A2 · A3 and A2 · A−2 are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in A · B.Our results are ‘local’ in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they ‘interact nicely’ with some other set.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/654da6316fea7923cd071e9476978f5728baea2a

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

The structure of approximate groups (2011, methodology): “Here results of multiplicative combinatorics, and in particular a lemma of Sanders [46] (see also [12]), are critical.”
Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. (2020, unknown): “This is achieved using an almost-periodicity result, which originated in the work of Croot and the second author [10], and the proof of which takes place entirely in physical space.”
Discrete Fourier analysis (2013, unknown): “1 with something like C(log 1/α) in place of the 1/α, using ideas on almost-periodicity from [4].”
The structure theory of set addition revisited (2012, methodology): “and then shortly after that Croot and Sisask came out with an important new argument in [CS10] which it turned out could be used to prove”
Higher moments of convolutions (2011, background): “Following Croot-Sisask argument we have P(X approximates A) > 1/2.”