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 †

What is the structure of a pair of finite integers sets A, B ⊂ Z with the small value of |A + B|? We answer this question for addition coefficient 3. The obtained theorem sharpens the corresponding results of G. Freiman.

Abstract. A brief historical introduction to the subject of additive combinatorics and a list of challenging open problems, most of which are contributed by the leading experts in the area, are presented.In this paper we collect assorted problems in additive combinatorics, including those which we qualify as classical, those contributed by our friends and colleagues, and those raised by the present authors. The paper is organized accordingly: after a historical survey (Section 1) we pass to the classical problems (Section 2), then proceed with the contributed problems (Sections 3-6), and conclude with the original problems (Section 7). Our problem collection is somewhat eclectic and by no means pretends to be complete; the number of problems can be easily doubled or tripled. We tried to include primarily those problems we came across in our research, or at least lying close to the area of our research interests. Additive combinatorics: a brief historical overviewAs the name suggests, additive combinatorics deals with combinatorial properties of algebraic objects, typically abelian groups, rings, or fields. That is, one is interested in those combinatorial properties of the set of elements of an algebraic structure, where the corresponding algebraic operation plays a crucial role. This subject is filled with many wondrous and deep theorems; the earliest of them is, perhaps, the basic CauchyDavenport theorem, proved in 1813 by Cauchy [16] and independently rediscovered in 1935 by Davenport [24,25]. This theorem says that if p is a prime, F p denotes the finite field with p elements (notation used throughout the rest of the paper), and the subsets A, B ⊆ F p are non-empty, then the sumset A + B := {a + b : a ∈ A, b ∈ B} has at least min{p, |A| + |B| − 1} elements. The analogue of this theorem for the set Z of integers is the almost immediate assertion (left as a simple exercise to the interested reader) that |A + B| ≥ |A| + |B| − 1 holds for any finite non-empty subsets A, B ⊆ Z.The F p -version of the problem is considerably more difficult, and all presently known proofs of the Cauchy-Davenport theorem incorporate a non-trivial idea, such as the transform method (sometimes called the "intersection-union trick"), the polynomial method, or Fourier analysis. The situation becomes even more complicated when one 1 2 ERNIE CROOT AND VSEVOLOD F. LEV considers subsets of a general abelian group. An extension of the Cauchy-Davenport theorem onto this case was provided by Kneser whose celebrated result [56,57] asserts that if A and B are finite, non-empty subsets of an abelian group with |A + B| < |A| + |B| − 1, then A + B is a union of cosets of a non-zero subgroup. Further refinement of Kneser's theorem was given by Kemperman in [55].Over a century passed between Cauchy's paper [16] and the next major result in the subject, proved by Schur [83] in the early 1900's. Schur's theorem states that for every fixed integer r > 0 and every r-coloring of the set N of natural numbers, there is a monochromatic triple (x, y, z) ∈ N...

We consider two general principles which allow us to reduce certain additive problems for residue classes modulo a prime to the corresponding problems for integers.