Abstract. We show that for integer n ≥ 1, any subset A ⊆ Z n 4 free of three-term arithmetic progressions has size |A| ≤ 4 γn , with an absolute constant γ ≈ 0.926. Background and MotivationIn his influential papers [R52, R53], Roth has shown that if a set A ⊆ {1, 2, . . . , N } does not contain three elements in an arithmetic progression, then |A| = o(N ) and indeed, |A| = O(N/ log log N ) as N grows. Since then, estimating the largest possible size of such a set has become one of the central problems in additive combinatorics. 4 / log N ). It is easily seen that Roth's problem is essentially equivalent to estimating the largest possible size of a subset of the cyclic group Z N , free of three-term arithmetic progressions. This makes it natural to investigate other finite abelian groups.We say that a subset A of an (additively written) abelian group G is progression-free if there do not exist pairwise distinct a, b, c ∈ A with a + b = 2c, and we denote by r 3 (G) the largest size of a progression-free subset A ⊆ G. For abelian groups G of odd order, Brown and Buhler [BB82] and independently Frankl, Graham, and Rödl [FGR87] proved that r 3 (G) = o(|G|) as |G| grows. Meshulam [M95], following the general lines of Roth's argument, has shown that if G is an abelian group of odd order, then r 3 (G) ≤ 2|G|/ rk(G) (where we use the standard notation rk(G) for the rank of G); in particular, r 3 (Z n m ) ≤ 2m n /n. Despite many efforts, no further progress was made for over 15 years, till Bateman and Katz in their ground-breaking paper [BK12] proved that r 3 (Z n 3 ) = O(3 n /n 1+ε ) with an absolute constant ε > 0.Abelian groups of even order were first considered in [L04] where, as a further elaboration on the Roth-Meshulam proof, it is shown that r 3 (G) < 2|G|/ rk(2G) for any finite abelian group G; here 2G = {2g : g ∈ G}. For the homocyclic groups of exponent 4 this †

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.

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 ) .

In the fastest-performing integer factoring algorithms, one creates a sequence of integers (in a pseudo-random way) and wishes to rapidly determine a subsequence whose product is a square. In 1994 Pomerance stated the following problem which encapsulates all of the key issues: Select integers a1, a2, . . . , at random from the interval [1, x], until some (nonempty ) subsequence has product equal to a square. Find a good estimate for the expected stopping time of this process. A good solution should allow one to determine the optimal choice of parameters in many factoring algorithms.Pomerance (1994), using an idea of Schroeppel (1985), showed that with probability 1 − o(1) the first subsequence whose product equals a square occurs after at least J 1−o(1) 0 integers have been selected, but no more than J0, for an appropriate (explicitly determined) J0 = J0(x). We tighten Pomerance's interval towhere γ = 0.577... is the Euler-Mascheroni constant, and believe that the correct interval is [(e −γ − o(1))J0, (e −γ + o(1))J0], a "sharp threshold". In our proof we confirm the well-established belief that, typically, none of the integers in the square product have large prime factors.The heart of the proof of our upper bound lies in delicate calculations in probabilistic graph theory, supported by comparative estimates on smooth numbers using precise information on saddle points.

In this paper we show that sumsets A + B of finite sets A and B of integers, must contain long arithmetic progressions. The methods we use are completely elementary, in contrast to other works, which often rely on harmonic analysis.