Open problems in additive combinatorics

Abstract

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...

Large sets in finite fields are sumsets

“…. , p−1 2 }, and every set of size p − 2 is an affine image of this S. Ben Green (see [4,5]) showed that if p is large then every subset S of Z p that consists of nearly all elements is a sumset. Let f (p) denote the maximum integer f so that every S ⊂ Z p of size at least p − f is a sumset.…”

From harmonic analysis to arithmetic combinatorics

“…For example, the region bounded by a three-cusped hypocycloid inscribed in a circle of radius 1 has the required property and has area π/8 ≈ . 39, whereas the area of the triangle is √ 3/3 ≈ 0.58. Kakeya's conjecture for the convex case was soon confirmed by Julius Pál [142], but the more interesting non-convex problem remained open.…”

“…Once more, we will focus on a small number of well-known problems representative of the area; for more comprehensive surveys see e.g. Croot-Lev [39], Granville [78], Nathanson [135], and Tao-Vu [187]. We begin with Freiman's theorem, a fundamental result on set addition.…”

“…A comprehensive up-to-date survey of additive combinatorics is in Tao-Vu [187]; general expository articles include Croot-Lev [39], Gowers [75], Granville [78], and Ruzsa [153]. Additive number theory, including Fourier-analytic methods, Freiman's theorem and inverse problems, is surveyed in Nathanson [135].…”

Roth’s theorem in ℤ4n

“…Indeed, for ‫ޚ‬ n 3 (or, more generally, any abelian group of odd order and bounded exponent), Roth's original argument simplifies considerably to give the following result, which is qualitatively due to Brown and Buhler [1984]. The question of what the true bounds on |A| are arises in many different studies [Frankl et al 1987;Yekhanin and Dumer 2004;Edel 2004;Edel et al 2007] and improving the bound is a well known open problem, as reported in [Green 2005;Croot and Lev 2007;Tao 2008, Section 3.1]; the closest anyone has come is in ]. While we are not able to make progress on this question, it is the purpose of this paper to show an improvement for a different class of groups.…”