A Probabilistic Technique for Finding Almost-periods in Additive Combinatorics
نویسندگان
چکیده
We introduce a new probabilistic technique for finding ‘almost-periods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on three-term arithmetic progressions and the existence of long arithmetic progressions in sumsets A+B in Zp. The bounds we obtain for these results are not the best ones known—these being established using Fourier analysis—but they are of a somewhat comparable quality, which is unusual for a method that is completely combinatorial. Furthermore, we are able to find long arithmetic progressions in sets A+B even when both A and B have density close to 1/ log p, which is much sparser than has previously been possible.
منابع مشابه
A Probabilistic Technique for Finding Almost-periods of Convolutions
We introduce a new probabilistic technique for finding ‘almost-periods’ of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type 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 ...
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