Abstract. 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 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 A 1 · A 2 · A 3 and A 2 · 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.
Given a positive integer N and real number α ∈ [0, 1], let m(α, N ) denote the minimum, over all sets A ⊆ Z N of size at least αN , of the normalized count of 3-term arithmetic progressions contained in A. A theorem of Croot states that m(α, N ) converges as N → ∞ through the primes, answering a question of Green. Using recent advances in higher-order Fourier analysis, we prove an extension of this theorem, showing that the result holds for k-term progressions for general k and further for all systems of integer linear forms of finite complexity. We also obtain a similar convergence result for the maximum densities of sets free of solutions to systems of linear equations. These results rely on a regularity method for functions on finite cyclic groups that we frame in terms of periodic nilsequences, using in particular some regularity results of Szegedy (relying on his joint work with Camarena) and the equidistribution results of Green and Tao.
Abstract.We prove results about the L p -almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L p , and gives a very short proof of a theorem of Green that if A and B are subsets of {1, . . . , N } of sizes αN and βN then A + B contains an arithmetic progression of length at least exp c(αβ log N ) 1/2 − log log N .Another almost-periodicity result improves this bound for densities decreasing with N : we show that under the above hypotheses the sumset A + B contains an arithmetic progression of length at least exp c α log N log 3 2β −1
1/2− log(β −1 log N ) .
We show that if A ⊂ {1, . . . , N } contains no non-trivial three-term arithmetic progressions then |A| ≪ N/(log N ) 1+c for some absolute constant c > 0. In particular, this proves the first non-trivial case of a conjecture of Erdős on arithmetic progressions.
Let α ∈ [0, 1] be a real number. Ernie Croot (Canad. Math. Bull. 51 (2008) 47–56) showed that the quantity maxA # (3‐term arithmetic progressions in A)/p2, where A ranges over all subsets of ℤ/pℤ of size at most α p, tends to a limit as p → ∞ through primes. Writing c(α) for this limit, we show that c(α)=α2/2 provided that α is smaller than some absolute constant. In fact, we prove rather more, establishing a structure theorem for sets having the maximal number of 3‐term progressions amongst all subsets of ℤ/p ℤ of cardinality m, provided that m < cp.
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