This book arose out of lecture notes developed by us while teaching courses on additive combinatorics at the University of California, Los Angeles and the University of California, San Diego. Additive combinatorics is currently a highly active area of research for several reasons, for example its many applications to additive number theory. One remarkable feature of the field is the use of tools from many diverse fields of mathematics, including elementary combinatorics, harmonic analysis, convex geometry, incidence geometry, graph theory, probability, algebraic geometry, and ergodic theory; this wealth of perspectives makes additive combinatorics a rich, fascinating, and multi-faceted subject. There are still many major problems left in the field, and it seems likely that many of these will require a combination of tools from several of the areas mentioned above in order to solve them.

The main purpose of this book is to gather all these diverse tools in one location, present them in a self-contained and introductory manner, and illustrate their application to problems in additive combinatorics. Many aspects of this material have already been covered in other papers and texts (and in particular several earlier books [168], [257], [116] have focused on some of the aspects of additive combinatorics), but this book attempts to present as many perspectives and techniques as possible in a unified setting.

Additive combinatorics is largely concerned with the additive structure1 of sets. To clarify what we mean by “additive structure”, let us introduce the following definitions.

In most of this book we have studied additive combinatorics problems in an ambient group Z, relying primarily on the additive structure of Z (as manifested for instance in the Fourier transform). However, in many cases the ambient group is in fact a field F, and thus supports a number of special functions, in particular polynomials. One can then use tools from algebraic geometry to exploit these polynomial structures; this is known as the polynomial method. One of the primary ideas here is to interpret an additive set (e.g. a sum set A + B) as the zero locus of one or more polynomials, possibly in several variables. One can then hope to control the size of such sets using results from algebraic geometry about the number and distribution of zeroes of polynomials. The most familiar example of such a theorem is the statement that a polynomial P(t) of one variable with degree d in a field F can have at most d zeroes; however for most applications we will need to study the zero locus of polynomial(s) in many variables. In this chapter we present four related tools and techniques from algebraic geometry which allow one to control such a zero locus. The first is the powerful combinatorial Nullstellensatz of Alon (Theorem 9.2), which asserts that the zero locus of a polynomial P(t1, …, tk) cannot contain a large box S1 × … × Sk if a certain monomial coefficient of P is non-vanishing; this is particularly useful for obtaining lower bounds on the size of restricted sum sets and similar objects.

In additive number theory, one frequently faces the problem of showing that a set A contains a subset B with a certain property P. A very powerful tool for such a problem is Erdős’ probabilistic method. In order to show that such a subset B exists, it suffices to prove that a properly defined random subset of A satisfies P with positive probability. The power of the probabilistic method has been justified by the fact that in most problems solved using this approach, it seems impossible to come up with a deterministically constructive proof of comparable simplicity.

In this chapter we are going to present several basic probabilistic tools together with some representative applications of the probabilistic method, particularly with regard to additive bases and the primes. We shall require several standard facts about the distribution of primes P = {2, 3, 5, …}; so as not to disrupt the flow of the chapter we have placed these facts in an appendix (Section 1.10).

Notation. We assume the existence of some sample space (usually this will be finite). If E is an event in this sample space, we use P(E) to denote the probability of E, and I(E) to denote the indicator function (thus I(E) = 1 if E occurs and 0 otherwise). If E, F are events, we use E ∧ F to denote the event that E, F both hold, E ∨ F to denote the event that at least one of E, F hold, and Ē to denote the event that E does not hold.

Introduction

One general theme throughout this book is that sum sets A + B are more structured than arbitrary sets A, B, and in particular that iterated sum sets such as lA = {a1 + … + al : ai ∈ Ai} should get increasingly structured as l gets larger. One example of this phenomenon is Lemma 4.13, which shows that if A has small Fourier bias then l A quickly fills out the entire ambient group. (See also Exercise 4.3.12 for related demonstration of special structure of sum sets.) For another example, let A be a subset of [1, n] for some large n and consider (as a measure of structure) the longest progression contained inside A. If A has no structure other than density, e.g. ∣A∣ ≥ 0.99n, then there is not much we can say. Even the powerful quantitative version (11.23) of Szemerédi's theorem due to Gowers can only obtain an arithmetic progression of length Ω(log log log log log n). For cubes the situation is somewhat better (and simpler); Lemma 10.49 guarantees that A contains a proper cube of dimension Ω(log log n), though this is still far from the maximal dimension Θ(log n) of the cubes inside [1, n].

The situation improves markedly with taking sum sets, though. First, if A ⊂ [1, n] has cardinality at least 0.99n, then it is easy to see that A + A or A − A contains an arithmetic progression of length 0.98n.

Incidence geometry deals with the incidences among basic geometrical objects such as points, lines and spheres. One can obtain useful and non-trivial information on these incidences by the classical combinatorial technique of double-counting the number of a certain type of configuration of incidences in two different ways.

In many situations, tools from from incidence geometry, combined with a clever double counting argument provide a simple, yet powerful, approach to hard problems. The goal of this chapter is to demonstrate several such applications, including several in additive combinatorics.

The material is organized as follows. We start with a result on the crossing number of graphs, which has a topological flavor. Next, we use this result to give simple proof of the famous Szemerédi–Trotter theorem concerning point-line incidences. In the next two sections, we use this theorem to prove several bounds on the Erdős–Szemerédi sum-product problem and reprove Andrew's theorem on the number of lattice points in a convex polygon. Next, we introduce the method of cell decomposition and use it to treat Erdős distinct distances problem in Rd. Finally, we discuss a variant of Erdős–Szemerédi sum-product problem for complex numbers.

The crossing number of a graph

In this chapter, a point refers to a point in the plane R2, and a line refers to a line in R2, unless otherwise specified. By a curve, we refer to the image of a continuous injective embedding of a compact interval [0, 1] into R2.

Consider a graphG = G(V, E); recall we assume our graphs G to be undirected and have no loops or repeated edges.

In Chapter 1 we have already seen the power of the probabilistic method in additive combinatorics, in which one understands the additive structure of a random object by means of computing various averages or moments of that object. In this chapter we develop an equally powerful tool, that of Fourier analysis. This is another way of computing averages and moments of additively structured objects; it is similar to the probabilistic method but with an important new ingredient, namely that the quantities being averaged are now “twisted” or “modulated” by some very special complex-valued phase functions known as characters. This gives rise to the concept of a Fourier coefficient of a set or function, which measures the bias that object has with respect to a certain character. These coefficients serve two major purposes in this theory. Firstly, one can exploit the orthogonality between different characters to obtain non-trivial bounds on these coefficients; this orthogonality plays a role somewhat similar to the role of independence in probability theory. Secondly, Fourier coefficients are very good at controlling the operation of convolution, which is the analog of the sum set operation, but for functions instead of sets. Because of this, the Fourier transform is ideal for studying certain arithmetic quantities, most notably the additive energy introduced in Definition 2.8.

Using Fourier analysis, one can essentially divide additive sets A into two classes. At one extreme are the pseudo-random sets, whose Fourier transform is very small (except at 0); we shall introduce the linear bias ∥A∥u and the Λ (p) constants to measure this pseudo-randomness.

In Chapter 2 we established the elementary theory of sum set estimates, showing how information on one sum A + B can be used to control other sums such as A − B or nA − mA. These estimates worked reasonably well even when the doubling constants of the sets involved were fairly large, since all the bounds were polynomial in this constant. On the other hand, we did not get detailed structural information on sets with small doubling constant; the best we could do is cover them by an approximate group (Proposition 2.26).

In this chapter we shall focus on the following question: given two additive sets A, B with A + B very small, what is the strongest structural statement one can then conclude about A and B? One of the main results in this area is Freiman's theorem which (in the torsion-free case) asserts that an additive set A with small doubling constant σ[A] = ∣2A∣/∣A∣ is contained in a progression of bounded rank which is not much larger than the original set. This theorem comes in a number of variants; we give several of them below. In doing so we shall also come across the useful concept of a Freiman homomorphism, which to a large extent frees the study of additive sets from the ambient group that they reside in, giving rise to a number of useful tricks, such as embedding the set inside a particularly nice group.

Additive combinatorics is a subfield of combinatorics, and so it is no surprise that graph theory plays an important role in this theory. Graph theory has already made an implicit appearance in previous chapters, most notably in the proof of the Balog–Szemerédi–Gowers theorem (Theorem 2.29). However there are several further ways in which graph theoretical tools can be utilized in additive combinatorics. We will only discuss a representative sample of these applications here. First we discuss Turán's theorem, which shows that sparse graphs contain large independent sets, and which is useful for constructing sum-free sets. Next we give a very brief tour of Ramsey theory, which allows one to find monochromatic structures in colored graphs (or other colored objects), in particular allowing one to find monochromatic progressions in any coloring of the integers (van der Waerden's theorem). Then we use some results about connectivity of dense graphs to establish the Balog–Szemerédi–Gowers theorem, which relates partial sum sets to complete sum sets and which has already been exploited in Chapter 2. Finally, we use the theory of commutative directed graphs to establish the Plünnecke inequalities, which are perhaps the sharpest inequalities known for sum sets and which strengthen several of the results already established in Chapter 2.

In Chapter 10 and Chapter 11 we shall discuss one final graph-theoretical tool, the Szemerédi regularity lemma, which has had many applications in several areas of discrete mathematics, but which in additive combinatorics has had an especially crucial role in the study of arithmetic progressions in dense sets.

Many classical problems in additive number theory revolve around the study of sum sets for specific sets A, B (though one typically works with infinite sets rather than finite ones). For instance, if N∧2 ≔ {0, 1, 4, 9, 16, …} is the set of square numbers, then it is a famous theorem of Lagrange that 4N∧2 = N, i.e. every natural number is the sum of four squares; if P ≔ {2, 3, 5, 7, 11, …} is the set of prime numbers, then it is a famous theorem of Vinogradov that (2 · N + 1)\3P is finite (i.e. every sufficiently large odd number is the sum of three primes); in fact it is conjectured that this exceptional set consists only of 1, 3, and 5. The corresponding result for (2 · N)\2P remains open; the infamous Goldbach conjecture asserts that 2P contains every even integer greater than 2, but this conjecture remains far from resolution.

In this text, we shall not focus on these types of problems, which rely heavily on the specific number-theoretic structure of the sets involved. Instead, we shall focus instead on the analysis of sum sets A + B and related objects for more general sets A, B.

In Chapter 2 we studied the elementary theory of sum sets A + B for general subsets A, B of an arbitrary additive group Z. In order to progress further with this theory, it is important first to understand an important subclass of such sets, namely those with a strong geometric and additive structure. Examples include (generalized) arithmetic progressions, convex sets, lattices, and finite subgroups. We will term the study of such sets (for want of a better name) additive geometry; this includes in particular the classical convex geometry of Minkowski (also known as geometry of numbers). Our aim here is to classify these sets and to understand the relationship between their geometrical structure, their dimension (or rank), their size (or volume, or measure), and their behavior under addition or subtraction. Despite looking rather different at first glance, it will transpire that progressions, lattices, groups, and convex bodies are all related to each other, both in a rigorous sense and also on the level of heuristic analogy. For instance, progressions and lattices play a similar role in arithmetic combinatorics that balls and subspaces play in the theory of normed vector spaces. In later sections, by combining methods of additive geometry, sum set estimates, Fourier analysis, and Freiman homomorphisms, we will be able to prove Freiman's theorem, which shows that all sets with small doubling constant can be efficiently approximated by progressions and similarly structured sets.

In this chapter we continue the study of Szemerédi's theorem (Theorem 10.1), but now focus on the case of longer arithmetic progressions k > 3. While we have seen that the k = 3 case of this theorem can be treated by Fourier-analytic methods, it turns out that the higher-k case cannot be dealt with by (linear) Fourier-analytic tools, even when k = 4; we will see some justifications for this fact later. Indeed, whereas Roth's treatment [287] of the k = 3 case appeared in 1953, it was only in 1969 that Szemerédi [343] established the k = 4 case of Theorem 10.1, by combining the density increment argument of Roth with some impressively complicated combinatorial arguments (based on those discussed in Section 10.7). Unfortunately, this argument did not yield any new bound on van der Waerden's theorem (Exercise 6.3.7), as this theorem was used in the proof; note that one of the original motivation of Erdős and Turán in introducing this problem in [99] was to obtain a more effective bound on van der Waerden's theorem than the Ackermann-type bounds obtained by the usual proof methods. In 1972 Roth [288] obtained an alternative proof of the k = 4 case by combining the Fourier method with Szemerédi's arguments, but again van der Waerden's theorem was involved.

In 1975, Szemerédi [345] finally established Theorem 10.1 for all k. The argument is purely combinatorial. It uses the density increment argument, van der Waerden's theorem, and an induction on k, although to execute this induction properly, a number of new combinatorial tools needed to be introduced, most notably the very useful and influential Szemerédi regularity lemma, which has already been discussed in Section 10.6.

Let υ1, …, υd be d elements of an additive group Z (which we refer to as the steps). Consider the 2d sums ∊1υ1 + … + ∊dυd with ∊1, …, ∊d ∈ {−1, 1}. In this chapter we investigate the largest possible repetitions among these sums.

We are going to consider two, opposite, problems:

• The Littlewood–Offord problem, which is to determine, given suitable non-degeneracy conditions on υ1, …, υd and Z (e.g. excluding the trivial case when all of the steps are zero), what the largest possible repetition or concentration can occur among these sums.

• The inverse Littlewood–Offord problem, which supposes as a hypothesis that the υ1, …, υd have a large number of repeated sums, or sums concentrating in a small set, and asks what one can then deduce as a consequence on the steps υ1, …, υd.

These two problems have a similar flavor to that of sum set estimates and inverse sum set estimates respectively, and occur naturally in certain problems of additive combinatorics, in particular in considering the set of subset sums FS(A) = {Σa∈Ba : B ⊂ A} of a given set A, or in the determinant and singularity properties of random matrices with entries ±1. These problems has also arisen in several other contexts, ranging from the zeroes of complex polynomials (which was the original motivation of Littlewood and Offord [237]), to database security (see [163]).