The Green-Tao Theorem on arithmetic progressions within the primes. Thomas Bloom

Transcription

1 The Green-Tao Theorem on arithmetic progressions within the primes Thomas Bloom November 7, 2010

2 Contents 1 Introduction Heuristics Arithmetic Progressions Structure of the Proof Arithmetic Progressions How to count arithmetic progressions Szemerédi Revisited Pseudorandomness Uniformity Norms and the Generalised von Neumann Theorem The Gowers Uniformity Norm The Generalised von Neumann Theorem Dual Norms Decomposition Theorem The Green-Tao Proof The Gowers-Hahn-Banach Proof The Relative Szemerédi Theorem Progressions in the Primes Counting the Primes and the W -trick Pseudorandom Majorant The Green-Tao Theorem Further Results Extensions of the Green-Tao Theorem Asymptotics for P (k, N) Explicit Bounds A Proof of the Decomposition Theorem 39 2

3 CONTENTS 3 B Estimates for Λ R 43 B.1 Euler Product for independent linear forms B.2 Euler product for simple linear forms B.3 Pseudorandomness of ν C Fourier transform 55 D The GI and MN Conjectures 57

4 4 Standard Notation The following notation and definitions are standard, and will be used without comment throughout this dissertation. A k-term arithmetic progression (or just a k-progression) is a set of the form {a + nb : 0 n k 1} for some a, b N. We exclude the degenerate case where b = 0. We say that f = O(g) if there exists a constant C such that for all sufficiently large x, f(x) Cg(x). f = o(1) if lim x f(x) = 0. f g if f = (1 + o(1))g. [n] := {1, 2,..., n 1, n}. φ(n) := #{m [n] : (m, n) = 1}. 1 if n = 1; µ(n) := ( 1) k if n is square-free and has k prime divisors; 0 otherwise. Specific notation and conventions We shall often be looking at the arithmetic mean of a function over a given set. For convenience, we denote this using the expectation notation, E(f(x) : x X) = E x X f(x) := 1 X f(x). Also, since we shall be trying to count k-term progressions of primes, it is convenient to introduce the following function: x X P (k, N) := # of k-term arithmetic progressions of primes in [N]. In many cases, we shall be taking an average over a hypercube; that is, points of the form (ω 1,..., ω n ) where ω i = 0 or 1 for all 1 i n. We denote the n-dimensional hypercube by C n := {0, 1} n. We denote the hypercube with the origin removed by C n := {0, 1} n {(0,..., 0)}. Where we are dealing with such points, of the form ω = (ω 1,..., ω n ) and h = (h 1,..., h n ), we define the scalar product to be ω h := ω 1 h ω n h n. We will be working mainly over the ring of integers modulo N, denoted by Z N := Z/NZ, and since we are letting N we may always assume that N is a prime, and so Z N is a field. We shall often be considering the space of functions f : Z N R, which we denote by R N. We give this the inner product f, g := E x ZN (f(x)g(x)),

5 CONTENTS 5 and where convenient the L p norms, for 1 p <, and f p := (E x ZN f(x) p ) 1/p f := sup x Z N f(x). We shall denote dependence on constants by subscripts. For example, O k,δ implies that the constant implicit in the O notation is dependent on the constants k and δ. Similarly, o k implies that the rate of decay is dependent on k. Since in most cases there is only one variable, I shall omit these subscripts for clarity, only making clear which variables the constants depend on when this is important or not clear from context. In almost all of this dissertation, the only variable is N, and hence, for instance, f = o(1) implies f tends to zero as N. The only other variable parameter is R := N, and hence we may still take the o(1) errors to be decaying as N. Whenever we use the variable p, we are ranging over primes. For example, p x p denotes the sum of all primes less than or equal to x. Acknowledgements I would like to thank my supervisor for his encouragement, advice and a careful reading of the first draft. I would also like to thank several of my fellow undergraduates for careful proofreading and comments on clarity and structure.

6 Chapter 1 Introduction Small arithmetic progressions within the primes are easy to come by. It is trivial to find progressions with one or two terms, and 3, 5, 7 gives a 3-term arithmetic progression. A moment s thought yields the 5-term arithmetic progression 5, 11, 17, 23, 29. The problem quickly becomes a lot more difficult the first 6-term arithmetic progression is 7, 37, 67, 97, 127, 157 and the current record holder has 26 terms: n for n = 0,..., A natural problem to pose is whether we can find such progressions within the primes for any given length. It is easy to prove that there can be no infinite arithmetic progression within the primes, so the best we can hope for is the following recent theorem of Ben Green and Terence Tao: Theorem 1.1 (Green-Tao, 2008 [10]). There are arbitrarily long arithmetic progressions within the primes. In fact, they prove something much stronger, and give an increasing function of N as a lower bound for how many such progressions are in the first N integers. It follows that there are in fact infinitely many arithmetic progressions of primes of any finite length. In this dissertation we describe a proof of this theorem, motivating and making clear the key steps and insights needed as we go along. It is a synthesis of methods and ideas from [10], [7] and [3], including some simplifications and new expository remarks. It is the first explicit description of the entire proof which includes the simplifications made since [10]. In this introductory section we present the background to the problem, including heuristics, related conjectures, and previous partial results which the Green-Tao theorem builds upon. We conclude by giving an overview of the structure of the proof. Chapters 2 to 5 focus on the different components of the proof, which are then brought together to prove (a stronger form of) Theorem 1.1 at the end of Chapter 5. Chapter 6 gives some extensions and related results which have since been obtained. 1 Found by Benot Perichon using the PrimeGrid software by Geoff Reynolds and Jaroslaw Wroblewski, April

7 CHAPTER 1. INTRODUCTION Heuristics The prime number theorem states that if π(x) is the number of primes less than or equal to x, then π(x) x log x. In probabilistic terms, this suggests that if we select an integer from [1, x] uniformly at random, then it is prime with probability roughly 1/ log x. This model fails as a method of proving statements about the primes if the primes were truly random then we would expect roughly about the same number of even and odd primes. The failure is because the model only considers the density of the primes, and not their arithmetical properties. This model is surprisingly useful, however, in formulating conjectures about how we expect the primes to behave. By including some information about their arithmetic properties (which often only changes the original conjecture by a constant) these heuristics can be converted into proven theorems. For example, consider the following problem: how many primes less than or equal to x are congruent to a modulo b? If a and b have a common divisor greater than 1, the answer is trivially either 0 or 1, so suppose a and b are coprime. Let us select integers below x uniformly at random. The probability that it is prime should be roughly 1/ log x and the probability that it is congruent to a modulo b is 1/φ(b), since there are φ(b) coprime congruence classes modulo b. Thus the probability that it meets both these conditions, assuming they are independent, should be 1/φ(b) log x and this leads us to conjecture that π a,b (x) x φ(b) log x where π a,b (x) is the number of primes less than or equal to x which are congruent to a modulo b. This conjecture turns out to be correct, and is known as the Prime Number Theorem for Arithmetic Progressions. Encouraged by the success of the probabilistic model in counting primes within a given arithmetic progression, let us try it with our problem of counting arithmetic progressions within the primes. We now use the expectation notation, together with the fact that if A is an event then E(# of times A occurs) = P(A): E(#a, b such that a, a + b,..., a + (k 1)b N are all prime) = P(a, a + b,..., a + (k 1)b are all prime) P(a is prime) P(a + (k 1)b is prime) 1 log k N where we have made the further assumption that the events of being prime are roughly independent. Furthermore, since there are (within a constant factor of) N 2 arithmetic

8 8 progressions of length k in [1, N], this leads us to conjecture that P (k, N) N 2 log k N. In particular, since the right hand side is unbounded as N tends to infinity, there are infinitely many k-term progressions within the primes. In fact, we shall prove something very similar to this asymptotic, giving a lower bound which only differs from the above heuristic by a constant factor. That is, we prove the following theorem. Theorem 1.2 (Green-Tao). For any k 3 there exists a constant c k > 0 such that, for all sufficiently large N, P (k, N) c k N 2 log k N. It is also possible to give an upper bound of this form, so P (k, N) is within a constant factor of the heuristic answer. The correct asymptotic seems to be similar - the heuristic above multiplied by some absolute constant, which reflects the arithmetical information about the primes which the probabilistic model does not include. Conjecture 1.1. For any k 3 there exists a constant C k > 0 depending only on k such that P (k, N) C k N 2 log k N. This has been verified for 3 k 6 see Chapter 6 for details. In fact, this is only a special case of a more general conjecture obtained by Hardy and Littlewood using similar heuristics. A linear form is a function in d variables of the shape where a i Q. a 0 + a 1 n 1 + a 2 n a d n d Conjecture 1.2 (Hardy-Littlewood Prime Tuples Conjecture [14]). Let ψ 1,..., ψ k be linear forms in d variables. Then for some constant C dependent only on the linear forms, #{n = (n 1,..., n d ) [0, N] d : ψ 1 (n),..., ψ k (n) prime} C log k N. An affirmative answer to this conjecture would not only give asymptotic for P (k, N), but also settle the long-standing Twin Primes Conjecture and Goldbach Conjecture. See [7] for more details. The Hardy-Littlewood conjecture is still unproven, although by extending the methods used in [10], Green and Tao have proven it for a significant class of linear forms in [7]. N d

9 CHAPTER 1. INTRODUCTION Arithmetic Progressions The advances which led to the Green-Tao theorem are more about the structure of arithmetic progressions than about the nature of the primes themselves. As Ben Green puts it in [5], they lie not in our understanding of the primes but rather in what we can say about arithmetic progressions. The problem of finding arithmetic progressions within sets began with a 1936 paper [1] of Erdős and Turán, in which they introduce the function r k (n), defined to be the size of the largest subset of {1,..., n} with no k-term arithmetic progression. The problem of locating k-term arithmetic progressions within sufficiently large sets is equivalent to showing that r k (n) grows sufficiently slowly. In particular, they conjectured that r k (n) = o(n) for any A {1,...,N} k. If we define the density of a set of integers A to be lim inf N, then this is N equivalent to the statement that any set of integers with positive density contains a k-term progression for any k. In fact, Erdős later made the stronger conjecture that Conjecture 1.3 (Erdős). If n A 1 n = then A contains arbitrarily long arithmetic progressions. ( This is essentially equivalent to showing that r k (n) = O n log n ) for any k. Note that this is stronger than r k (n) = o(n), since it gives an explicit upper bound on the rate of decay. The Green-Tao theorem would be an immediate corollary of this conjecture, since 1 p diverges. p The weaker conjecture that r k (n) = o(n) was proven for the first non-trivial case k = 3 by Roth in 1956, giving the following theorem. Theorem 1.3 (Roth, 1956 [18]). Any subset of N with positive density contains a 3-term arithmetic progression. The case k = 4 was proven by Szemerédi in 1969, and in 1975 he managed to extend this to arbitrary k using complicated combinatorial arguments. Theorem 1.4 (Szemerédi, 1975 [19]). Any subset of N with positive density contains arbitrarily long arithmetic progressions. In fact, we can strengthen Szemerédi s theorem to show that we can find not just one, but infinitely many progressions of any finite length using a simple combinatorial argument first noted by Varnavides. Corollary 1.1 (Varnavides, 1959 [24]). Let A N and k 2. If there exists δ > 0 such that A {1,..., N} δ for all sufficiently large N, then there exists a constant c k,δ > 0 such that, for sufficiently large N, there are at least c k,δ N 2 k-term arithmetic progressions in A {1,..., N}.

10 10 The prime number theorem implies the primes have density zero, and hence Szemerédi s theorem cannot be applied directly. It will, however, be invoked at a crucial step in proving the Green-Tao theorem. Szemerédi s theorem has been reproven several times since 1975 using methods from a surprisingly diverse array of mathematical fields first by Furstenberg in 1977 using ergodic theory and then twice by Gowers, using Fourier analysis and hypergraph regularity. More recently, in 2009 an alternative combinatorial proof was found by the internet polymath project. For a detailed survey of the different proofs see [21]. It was insights gained from studying the common features of these different proofs which led to the methods used by Green and Tao. The first progress on the problem of progressions within the primes came from van der Corput in 1939, who settled the result for k = 3 using the circle method. Theorem 1.5 (van der Corput, 1939 [23]). There are infinitely many arithmetic progressions consisting of three primes. After this theorem, although significant results were obtained for sets of positive density as outlined above, no further results were obtained for the primes until 1981 when Heath- Brown showed, using methods similar to van der Corput s, the following partial result for k = 4. Theorem 1.6 (Heath-Brown, 1981 [15]). There are infinitely many arithmetic progressions consisting of three primes and one almost-prime (that is, a number with only two prime factors, counted with multiplicity). 1.3 Structure of the Proof The outline of the proof given here is new, although of course all the ideas are implicit in the original approach of Green and Tao. In [10], however, they did not make the transference principle explicit; it is discussed in more detail in, for example, [3]. The fundamental insight behind Green and Tao s work was that, heuristically, a large random subset of the integers is very similar to the integers themselves, conclusions which hold for the latter should hold for the former. In other words, there should be a kind of transference principle which would allow results which hold for the integers to hold for sufficiently random subsets. Let us call such subsets pseudorandom sets. Applying the transference principle to Szemerédi s theorem, we may hope the following to hold. Conjecture 1.4 (Relative Szemerédi Theorem). Let X N be sufficiently pseudorandom. Then any subset of X with positive density inside X has arbitrarily long arithmetic progressions. In particular, although the primes have zero density within N, we may hope to find some pseudorandom set X N in which the primes have positive density, and deduce that the

11 CHAPTER 1. INTRODUCTION 11 primes contain arbitrarily long arithmetic progressions. In [10], Green and Tao do exactly this, and their proof can be divided into two distinct parts. The first is proving the Relative Szemerédi theorem that is, showing that the kind of structure reflected in Szemerédi s theorem is amenable to the transference principle mentioned above. This is accomplished using the machinery of Gowers uniformity norms, first introduced by Gowers to prove Szemerédi s theorem. In particular, these norms induce a notion of distance between subsets of N with the following properties. The first is that this distance preserves the structure of containing arithmetic progressions, in that sets which are close have roughly the same number of arithmetic progressions. This is formalised as the Generalised von Neumann theorem. The mathematics needed here relies on the Gowers uniformity norms, and is similar to the kind of regularity arguments used in the hypergraph proof of Szemerédi s theorem by Gowers. This, along with a discussion of the uniformity norms, is the subject of Chapter 3. The second is the transference principle mentioned above: a set which is dense inside some pseudorandom subset of the integers is close to a set which is dense within the integers. This is formalised as the Decomposition theorem. In the original proof, Green and Tao used finitary ergodic theory to prove this, inspired by Furstenberg s proof of Szemerédi s theorem using ergodic theory. In this dissertation, we present a simpler proof discovered by Gowers using functional analysis. This is the subject of Chapter 4. The third component is Szemerédi s theorem, which shows that the larger set obtained from the decomposition contains many arithmetic progressions. We then invoke the Generalised von Neumann theorem to show that the original set also contains many arithmetic progressions. This finishes the proof of the Relative Szemerédi theorem. With this in place, the second part of the proof of the Green-Tao theorem is showing that the hypothesis of the Relative Szemerédi theorem is met: the primes sit inside a pseudorandom set with positive density. For this, we will use a weighted version of the almost-primes numbers with few prime factors. We will discuss this part of the proof in Chapter 5. To show that the almost-primes are sufficiently pseudorandom uses techniques from traditional analytic number theory, and Green and Tao were able to use arguments and results already established by Goldston and Yıldırım in their work on small gaps between the primes [2]. This part of the argument has since been simplified; we have incorporated these simplifications into Chapter 5.

12 Chapter 2 Arithmetic Progressions In this chapter we discuss arithmetic progressions and Szemerédi s theorem in more detail, and formulate the precise definition of pseudorandomness which we shall require. The exposition in this chapter is new, but the ideas it discusses are all present in [10] and earlier work. 2.1 How to count arithmetic progressions In many problems dealing with the existence of certain structures in the natural numbers, it is easier to try to solve the apparently more difficult problem of counting how many such structures we may expect to find in any finite subset of the natural numbers. Hence it is sufficient to show that this count is not zero. Another simplification that can be made is to consider functions instead of sets. We may pass from considering a subset A N to its characteristic function 1 A : N {0, 1} using the following important observation: { 1 if x, x + r,..., x + (k 1)r A; 1 A (x)1 A (x + r) 1 A (x + (k 1)r) = 0 otherwise. Hence we may count all arithmetic progressions within A Z N by the sum x,r Z N 1 A (x)1 A (x + r) 1 A (x + (k 1)r). Note that we have switched from considering arithmetic progressions within {1,..., N} to those within Z N. This is to avoid the problem that, for instance, x, r {1,..., N} does not guarantee that x + (k 1)r {1,..., N}. We discuss this issue further below. Statements about the existence of arithmetic progressions then reduce to the above sum being non-zero. In fact, whenever the sum is bounded below by a constant, it is also bounded below by a constant multiple of N 2 for sufficiently large N, thanks to simple arguments such as were used by Varnavides to prove Corollary 1.1. Hence we in fact consider the above sum 12

13 CHAPTER 2. ARITHMETIC PROGRESSIONS 13 weighted by 1 N 2. This leads us to the expectation notation, and motivates us to make the following definition: Definition 2.1. Let f 0,..., f k 1 : Z N R. The normalised count of k-term arithmetic progressions in f 0,..., f k 1 is defined by Υ k (f 0,..., f k 1 ) := E (f 0 (x)f 1 (x + r) f k 1 (x + (k 1)r) : x, r Z N ). The normalised count of k-term arithmetic progressions in f is Υ k (f) := Υ k (f,..., f). Remark 2.1. The use of Υ k was not present in [10], although they repeatedly use the expectation it is shorthand for. Conventionally this expectation is denoted by Λ, but in this context this would create confusion with the von Mangoldt function for the primes. The discussion above shows that, for A Z N, there are N 2 Υ k (1 A ) many k-term arithmetic progressions in A. There are two important things to observe about the way we are counting arithmetic progressions. The first is that we include the degenerate case when r = 0. This will not be a problem, as such degenerate cases will contribute at most 1/N to Υ k, which we shall only be estimating up to o(1) errors. The second potential problem is the wraparound issue noted above we are counting arithmetic progressions in Z N, rather than {1,..., N}. For instance, when N = 5, we would include {1, 4, 2} as a 3-term arithmetic progression. This will happen if and only if, for some 1 i k 1, the term a + ir is larger than N, for then it would have to wraparound Z N. One crude way to avoid this, which will be sufficient, is to restrict a and r to being less than N/k. In the case of the primes, we will ensure this by incorporating some small factor in the definition of our counting function f to ensure such wraparound arithmetic progressions are not counted. 2.2 Szemerédi Revisited By considering characteristic functions instead of sets, using a Varnavides argument, and using the wraparound trick mentioned above to pass from considering {1,..., N} to Z N, we may rewrite Theorem 1.4 as follows: Theorem 2.1. For any k 1 and δ > 0 there exists a constant c k,δ > 0 such that the following holds. Let f : Z N {0, 1} such that, for sufficiently large N, the density Ef δ. Then for sufficiently large N, Υ k (f) c k,δ. An important consequence of the approach of Gowers was the realisation that the function f need not be discrete, but it is sufficient that it be bounded above by 1. This leads us to the following formulation of Szemerédi s theorem:

14 14 Theorem 2.2. For any k 1 and δ > 0 there exists a constant c k,δ > 0 such that the following holds. Let f : Z N R such that 0 f(x) 1 for all x Z N and, for sufficiently large N, the density Ef δ. Then for sufficiently large N, Υ k (f) c k,δ. As explained in the introduction, this theorem solves the problem adequately for sufficiently dense sets of integers, but cannot be applied to the primes, since they have zero density. In particular, if we let 1 P be the characteristic function of the primes, then the prime number theorem implies that E1 P 1 0 as N. Thus the Ef δ > 0 log N hypothesis of Theorem 2.2 is not satisfied. We can avoid this and increase the density of our prime counting function by weighting it with a log N factor that is, we instead consider the function f := log N 1 P. This now satisfies the density hypothesis, but is no longer bounded above by 1. The Relative Szemerédi theorem allows us to weaken this restriction, requiring only that it is bounded above by some sufficiently pseudorandom function. In particular, as an analogue to Theorem 2.2, we have the following precise version of Conjecture 1.4. Conjecture 2.1 (Relative Szemerédi Theorem). Let ν : Z N R be k-pseudorandom, and let f : Z N R such that 0 f ν. If there exists a constant δ > 0 such that, for sufficiently large N, the density Ef δ, then there exists a constant c k,δ > 0 such that, for sufficiently large N, Υ k (f) c k,δ. To prove this theorem, we use the strategy outlined in the introduction. It will be proven in Chapter 4 as Theorem Pseudorandomness This section gives some original remarks on the notion of pseudorandomness from [10], and presents a clearer classification of the pseudorandomness condition into two components. We first explain what kind of pseudorandomness we will need. Recall that ν is a function from Z N to R, and should be thought of as a weighted indicator function for a subset of the integers which is sufficiently pseudorandom for a transference principle to hold. The precise conditions given below are determined by what we require in the technical theorems to come later, but they reflect the following principles: 1. ν should behave like the constant function 1, since the pseudorandom set we transfer to should behave like the integers. 2. The events ν(a) and ν(b) should be independent, for distinct a and b, since the probability that distinct elements belong to the pseudorandom set is independent. We divide the definition of pseudorandom below into two parts, which correspond to the two components of the Relative Szemerédi theorem: the Generalised von Neumann theorem

15 CHAPTER 2. ARITHMETIC PROGRESSIONS 15 and the Decomposition theorem. In [10], Green and Tao also divide up the definition into two parts, though they do it differently into a linear forms condition and a correlation condition. In that form, however, it is not clear that there is a distinction between the types of pseudorandomness which the two components require. The first is the pseudorandomness required to prove the Generalised von Neumann theorem. Definition 2.2 (Linear Pseudorandomness). We say that ν is linearly k-pseudorandom if whenever we have a system of m k2 k 1 linear forms in t 3k 4 variables, ψ i (x) := t L ij x j where 1 i m, j=1 such that none of the t-tuples (L ij ) 1 j t Q t are zero, none is a rational multiple of another, and moreover for each i, j the coefficient L ij Q has numerator and denominator bounded by k in absolute value, then E x Z t N (ν(ψ 1 (x)) ν(ψ m (x))) = 1 + o k (1). Remark 2.2. All the linear forms in this condition are assumed to be homogenous, that is, having zero constant term so ψ(0) = 0. In particular, we have the measure condition, E(ν) = 1 + o(1), which agrees with our first principle. Linear pseudorandomness should be viewed as a kind of independence between ν(ψ 1 ),..., ν(ψ m ), in accordance with our second principle. This is a very strong condition, since it gives us a great deal of control over a large class of linear forms, in particular the k linear forms in 2 variables which give a k-term arithmetic progression: ψ 1 (x 1, x 2 ) := x 1, ψ 2 (x 1, x 2 ) := x 1 + x 2,..., ψ k (x 1, x 2 ) := x 1 + (k 1)x 2. From the linear pseudorandomness condition with these linear forms we get Υ k (ν) = 1 + o(1), and a lot of arithmetic progressions counted by our pseudorandom function. The power of the transference principle is that we don t lose too many of these when passing to suitable f ν. The next condition is required for the Decomposition theorem to hold. Definition 2.3 (Simple Pseudorandomness). We say that ν is simply k-pseudorandom if whenever we have m 2 k 1 simple linear forms ψ i in t k variables, that is, ones of the shape t ψ i (x) := ω ij x j + b i j=1

16 16 where ω ij {0, 1}, such that the affine parts ω i = (ω i1,..., ω it ) are not zero or rational multiples of each other, then E(ν(ψ 1 (x)) ν(ψ m (x)) = 1 + o(1). Furthermore, there exists a weight function τ m : Z N R + such that E(τ q ) = O m,q (1) for all 1 q < and for all h 1,..., h m Z N we have the upper bound E x ZN (ν(x + h 1 ) ν(x + h m )) τ(h i h j ). 1 i<j m Remark 2.3. We cannot apply the first part to give an asymptotic for the second part, since the affine parts of the linear forms are all the same. We also observe that we cannot control these expressions using linear pseudorandomness, since the forms are non-homogenous. We now make the following umbrella definition, which is required for the Relative Szemerédi theorem. Definition 2.4 (Pseudorandomness). We say that ν is k-pseudorandom if it is both linearly k-pseudorandom and simply k-pseudorandom. The constant function 1 is the easiest example of a pseudorandom function. In fact, it is also an important one, since the space of pseudorandom functions is star-shaped around 1, as the following easily verified lemma shows. Lemma 2.1. If ν is linearly pseudorandom, then so is λν + (1 λ) for any λ (0, 1); similarly for simple pseudorandomness. This lemma will be important in several places, since it will allow us to pass from bounds of the form f ν + 1 to ones of form f ν losing only a constant factor, but preserving pseudorandomness. Remark 2.4. It is believed that these conditions are stronger than necessary. Weakening the strength of the pseudorandomness necessary (particular the simple pseudorandomness required for the Decomposition theorem) is one goal of current research in this area.

17 Chapter 3 Uniformity Norms and the Generalised von Neumann Theorem In this chapter we introduce the Gowers uniformity norm, which will play a central role in the proof. We also prove the first component of the Relative Szemerédi theorem, the Generalised von Neumann theorem. Once again, the substantial ideas in this chapter are all present in [10], though the exposition in the first section contains some new ideas. 3.1 The Gowers Uniformity Norm Recall that our strategy for proving the Relative Szemerédi theorem is to show that a set dense in a pseudorandom set is close in some sense to one dense in the natural numbers. In terms of the functional approach in the previous chapter, we seek some metric d on R N such that: 1. If d(f, g) is small then f and g count a similar number of k-term arithmetic progressions, and 2. If f ν for some pseudorandom ν then there exists a bounded g such that d(f, g) is small. The easiest way to obtain a metric is to induce it from some norm on the space. We now give the definition of the required norm as in [10]. First, however, we give some original remarks to help motivate the definition. We seek to decompose a function f ν, where ν is a pseudorandom function, as f = g+h where g is bounded and h is small, in the sense that Υ k (g + h) is well approximated by Υ k (g). We now need to specify what is meant by small. Our initial approach might be to use Υ k, the normalised count of arithmetic progressions, directly that is, we hope to achieve a decomposition where Υ k (h) is small. There are two problems with this approach. 17

18 18 The first is that we hope to approximate Υ k (g + h) by Υ k (g), and so we need that Υ k (g + h) = Υ k (g) + Υ k (f 1,..., f k ) =I [k] = Υ k (g) + negligible terms. where f i = h if i I and f i = g otherwise. In other words, we need not only Υ k (h) small, but also Υ k (f 1,..., f k ) small whenever some f i = h. It would be sufficient to prove an inequality such as Υ k (f 1,..., f k ) = O k ( inf Υ k(f i )). 1 i k This cannot hold, however, as shown by the following counterexample. Define f 1 (n) = 1 if n = 0, and f 1 (n) = 0 otherwise, and let f i 1 for all i 2. Then inf Υ k (f i ) = Υ k (f 1 ) = 1/N 2, whereas Υ k (f 1,..., f k ) = 1/N for all N. The second problem with using Υ k directly is that it is not a norm on R N, for the trivial reason that it can be negative. We may avoid this by taking the absolute value, but although it is easily verified that Υ k is a seminorm, it is not a norm. For example, take N = 3, and f : Z 3 R defined by f(0) = 0, f(1) = 1 and f(2) = 1. A simple calculation shows Υ k (f) = 0, although f 0. This is a problem for the existence of a decomposition, since the analytic machinery we hope to use to find such a decomposition relies on h being small in some norm on R N. Hence we should not demand that h be small in terms of Υ k, but rather in some other norm on R N. To discover what this should look like, focus on the first of the problems above bounding Υ k (f 1,..., f k ). The most common tool in bounding expectations is the Cauchy-Schwarz inequality, E(XY ) 2 E(X 2 )E(Y 2 ). We need to bound Υ k in terms of something involving only h, and so we must remove k 1 functions. For concreteness, let us temporarily fix k = 3. After applying the Cauchy-Schwarz inequality twice, we may bound the Υ 3 (f 1, f 2, f 3 ) term, E(f 1 (x)f 1 (x + r)f 2 (x + r + r) : x, r Z N ), with a product where each term has the shape E(f(x)f(x + h 1 )f(x + h 2 )f(x + h 1 + h 2 ) : x, h 1, h 2 Z N ). This is similar to the shape of Υ 3, but with the sum r+r replaced with h 1 +h 2 for independent variables h 1, h 2. This suggests that if h is small with respect to this expectation, then we can use the Cauchy-Schwarz inequality to show that Υ 3 (f 1, f 2, f 3 ) is small whenever some f i = h, and hence that Υ 3 (g + h) Υ 3 (g). Motivated by this, we make the following definition. Definition 3.1. For any d 1, the Gowers d-uniformity norm of a function f : Z N R is defined as ( ) 1/2 d f U d := E f(x + ω h) : x Z N, h Z d N. ω C d

19 CHAPTER 3. UNIFORMITY NORMS AND THE GENERALISED VON NEUMANN THEOREM 19 Note that, in the k = 3 case, f 4 U is exactly the expectation we obtained above. In 2 general, we will use the U k 1 norm to deal with progressions of length k. It is easy to verify that U d is a seminorm for d 1. That it is also a genuine norm when d 2 follows from the easily verified fact that f U 2 = ˆf 4 where ˆf is the Fourier transform of f, and the less obvious monotonicity property f U d 1 f U d. Recalling that we are seeking an inequality of the shape # of k-progressions counted by f f U k 1, the monotonicity property agrees with the trivial observation that any (k + 1)-progression truncated gives a k-progression (and hence if the count of (k + 1)-progressions is small, then so is the count of k-progressions). With this definition in place, we can restate our strategy for proving the Relative Szemerédi theorem. We need to show that the U k 1 norm has the following properties. 1. If h U k 1 is small then (for suitable g) Υ k (g) Υ k (g + h). 2. If f ν then f = g + h where g is bounded and h U k 1 is small. The first is the Generalised von Neumann Theorem, which occupies the next section. The second is the crucial Decomposition Theorem, which we discuss in the next chapter. We need to control the count of arithmetic progressions over functions with small Gowers uniformity norm. Since we shall often be referring to functions with small Gowers uniformity norms, it is convenient to make the following definition. Definition 3.2. We say that f is η-uniform if f U d η, and more generally say that f is uniform if f U d is small. For a more in-depth discussion of the Gowers uniformity norms, including a proof of the monotonicity property mentioned above, see (for example) Appendix B of [7]. 3.2 The Generalised von Neumann Theorem We come now to the first component needed to prove the Relative Szemerédi theorem. A specialised form of this theorem, when ν 1, was first used by Gowers in his proof of Szemerédi s theorem. The fact that it could be generalised to linearly pseudorandom ν was first noticed by Green and Tao in [10] indeed, the linearly pseudorandom condition which we require ν to satisfy was chosen with the proof of this theorem in mind. The proof is long and technical, and can be found in [10]. The idea is to repeatedly apply the Cauchy-Schwarz inequality as outlined above until we are at a stage where we can apply the pseudorandom condition.

20 20 Theorem 3.1 (Generalised von Neumann Theorem). Let ν be linearly pseudorandom, and f 0,..., f k 1 obey the bounds f i (x) ν(x) for all x. Then Υ k (f 0,..., f k 1 ) = O k ( inf f i U k 1) + o k (1) 0 i k 1 Remark 3.1. Using Theorem 2.1, and rescaling the f i where necessary, we may in fact weaken the conditions to f i ν + 2. This fact will be needed when we apply this to prove the Relative Szemerédi theorem, since we will need to apply it to h = f g where 0 f ν and 0 g 2. Proof. Omitted. See [10], Section 3. In particular, if h is uniform, then Υ k (g + h) is approximately Υ k (g). This is the key step in the proof of the Relative Szemerédi theorem, so we formulate it precisely as follows. Corollary 3.1. If f = g + h where g, h ν for some linearly pseudorandom ν, and h is η-uniform, then Υ k (f) = Υ k (g) + O k (η) + o(1). Proof. Expanding out the expectation notation, we see that Υ k (g + h) = Υ k (g) + Υ(f 1,..., f k ) =I [k] where f i = h if i I and f i = g otherwise. We then apply Theorem 3.1 and the condition that h is η-uniform to show that each of the terms in the sum is bounded by O k (η)+o k (1). 3.3 Dual Norms This section closely follows the first part of section 6 in [10], although due to the usefulness of dual norms in the new approach to the Decomposition theorem in the next chapter, we define the dual norm in generality. In general, whenever we have a norm on R N we may define the dual norm as follows: f := sup{ f, g : g 1}. It is easy to check that this defines a seminorm on R N, and for the norms we shall be dealing with it will also be a norm. The use of this definition lies in the inequality f, g f g. In particular, whenever g is small, and g correlates with f to a large degree, then f must be large. That is, smallness of the dual norm prevents the norm of related functions from being small. In the case of the U d norms, we say that g is anti-uniform if it has small dual U d norm, and so anti-uniformity is an obstruction to uniformity. Closely linked to the introduction of dual norms, we also introduce the concept of dual functions at least, with respect to the U d norms. In the following definition, and throughout

21 CHAPTER 3. UNIFORMITY NORMS AND THE GENERALISED VON NEUMANN THEOREM 21 the rest of this dissertation, we shall fix d = k 1, recalling that k is to be taken as a fixed quantity. The dual function of f is defined as Df(x) := E f(x + ω h) : h Z k 1. ω C k 1 0 The use of this lies in the following lemma. This will be useful later, when we shall apply it to deduce that sufficiently uniform functions do not correlate much with their dual functions. Lemma 3.1. f, Df = f 2k 1 U k 1. Proof. Expand out both sides using their definitions. N

22 Chapter 4 Decomposition Theorem The goal of this chapter is to prove the following. Theorem 4.1 (Decomposition Theorem). Let ν be simply pseudorandom, and η some parameter such that 1 > η > 0. Suppose N is sufficiently large, depending on η. Then for every function 0 f ν we can decompose it as f = g + h where 0 g 2 and h is η-uniform. This is the final, and most crucial part of the proof of the Relative Szemerédi theorem, and hence of the entire Green-Tao theorem. It is presented as a decomposition, which allows us to decompose f into a bounded part (to which we may apply Szemerédi s theorem) and a uniform part, whose contribution is negligible by the Generalised von Neumann theorem. It is, however, better viewed as a transference theorem: it allows us to transfer properties of the integers to pseudorandom subsets of the integers. In this case, the desired property is that dense subsets contain arbitrarily long arithmetic progressions. The relationship between decomposition theorems and transference theorems holds in quite general terms, and is discussed in detail in [3]. 4.1 The Green-Tao Proof The original proof used by Green and Tao in [10] is quite different to the one present below, and relies on a finitary ergodic theory inspired by Furstenberg s proof of Szemerédi s theorem. We briefly sketch their approach here before presenting the simpler proof by Gowers. Their proof constructs the decomposition in stages. They begin by looking at the decomposition f = E(f) + (f E(f)). It follows from the pseudorandomness of ν that E(f) is bounded, so the remaining problem is to show that f E(f) is sufficiently uniform. Of course, there is no guarantee that it will be. Instead, they use the machinery of conditional expectations over σ-algebras to increase the uniformity as follows. By using dual functions as obstructions to uniformity, if f E(f) is not sufficiently uniform sets can be added to create an expanded σ-algebra B. These new sets are chosen so that the conditional expectation E(f B) absorbs the impact of the dual functions which were obstructing the 22

23 CHAPTER 4. DECOMPOSITION THEOREM 23 uniformity. In particular, the difference f E(f B) lacks these obstructions, and is more uniform. Furthermore, it follows from the pseudorandomness of ν and the fact that f ν that E(f B) remains bounded. They continue in this fashion, keeping E(f B) bounded at each stage, while increasing the uniformity of f E(f B). There is no guarantee, however, that this process will terminate that is, while the approximations are becoming more uniform at each stage, they may never become sufficiently uniform. Green and Tao show that this process must terminate using an energy increment argument used in several approaches to Szemerédi s theorem. This argument uses the fact that at each stage in their construction, the pseudorandomness of ν ensures that E(f B) remains bounded. The energy, that is, the L 2 -norm, of E(f B) increases at each stage, but since it is bounded above, there must be a stage at which the energy may not increase, and hence no further approximations can be made and the process must terminate. If the process has terminated, however, it must mean that the approximation at that stage was sufficiently uniform, and so the decomposition at this stage meets our requirements. 4.2 The Gowers-Hahn-Banach Proof The simpler proof outlined in this section takes a very different approach. Rather than constructing a decomposition explicitly, it uses the Hahn-Banach theorem to derive a contradiction if no decomposition exists. This approach was independently discovered by Gowers [3] and Reingold, Trevisan, Tulsiani and Vadhan [17]. The proof we give here follows the outline given in [3]. Some parts of the argument have been simplified, since we do not require the generality given by Gowers, and the presentation of the argument given here is new. We begin by stating the version of the Hahn-Banach theorem 1 that we will use. Theorem 4.2 (Hahn-Banach theorem). Let K 1 and K 2 be closed convex subsets of R N, each containing 0, and suppose that f R N cannot be written as a convex combination c 1 f 1 + c 2 f 2 with f i K i. Then there exists φ R N such that f, φ > 1 and g, φ 1 for every g K 1 K 2. With this theorem available to us, the strategy should be fairly obvious. Recall that we need a decomposition f = g + h where g is bounded and h is uniform. We suppose that no such decomposition exists and use Theorem 4.2 to derive a contradiction. Roughly, this will be as follows: f, φ will be large, but ν, φ will be small, contradicting the fact that f ν. We hope to say that ν, φ is small since it is the sum of 1, φ, which is bounded, and ν 1, φ, which is o(1) since ν 1 is uniform and φ is anti-uniform. There are, however, significant technical difficulties to be overcome before we can put this into action. First, we need the following simple consequence of pseudorandomness. Lemma 4.1 (Uniformity of ν 1). If ν : Z N R is simply k-pseudorandom, then ν 1 U k 1 = o(1). 1 This is quite different from the Hahn-Banach theorem as it is usually stated; for a derivation of the version stated, see [3].

24 24 Proof sketch. Expand out the definitions and use the binomial theorem. Now let us try to prove Theorem 4.1 using only this Lemma and the Hahn-Banach theorem. In terms of the latter, we have two closed convex subsets of R N : positive functions bounded by 2 and functions which are η-uniform. If the decomposition does not hold, then by Theorem 4.2 we can find some function φ such that 1. f, φ > 1, 2. g, φ 1 for every g such that 0 g 2, and 3. h, φ 1 for every h such that h U k 1 η. In particular, by setting g to be the function g(x) = 2 whenever φ(x) 0 and g(x) = 0 otherwise, we can suppose that 1, φ + 1 2, where φ + is the positive part of φ defined by φ + (x) := φ(x) when φ(x) 0 and 0 otherwise. We have the following chain of inequalities: 1 < f, φ f, φ + ν, φ + = 1, φ + + ν 1, φ φ + U k 1 ν 1 U k 1. Using Lemma 4.1, to obtain a contradiction for N sufficiently large it suffices to show that φ + is anti-uniform. The problem is that condition 3 only gives us a bound for φ U k 1, and this is not strong enough. The difficulty lies in passing from φ from φ +, which is necessary since we can only deduce from f ν that f, φ ν, φ if φ is strictly non-negative; if some stronger version of Theorem 4.2 were available that guaranteed φ 0 then the simple argument given above would be sufficient. In particular, instead of simple pseudorandomness, all we would need is the weaker condition ν 1 U k 1 = o(1). To fix this argument, we will show that φ + can be approximated by a function that is anti-uniform. This is technically messy, and we leave the details to Appendix A. It is in proving this approximation, however, that the majority of the simple pseudorandomness condition is required. It gives the following theorem. Theorem 4.3 (Approximation with an anti-uniform function). Condition (3) above implies that there exists a function ψ such that ψ φ + 1/8 and ψ A for some A U k 1 depending only on η. Using this theorem, we may adapt the chain of inequalities above to use this approximation to φ +, and obtain the inequality 1 < f, φ o(1) + A ν 1 U k 1. Since A is fixed and ν 1 U k 1 is o(1), we have a contradiction for N large enough, which proves Theorem 4.1. Once again, the details are technical and left to Appendix A.

25 CHAPTER 4. DECOMPOSITION THEOREM The Relative Szemerédi Theorem We now have all we need to prove the main component of the Green-Tao theorem. The proof below fills in the sketch given in [10], making some minor changes since our Decomposition theorem is different to the form in which it is given there. Theorem 4.4 (Relative Szemerédi Theorem). Let k 3 and δ > 0, and let ν : Z N R + be k-pseudorandom. Suppose f : Z N R satisfies 0 f(x) ν(x) for all x Z N and Ef δ. Then for all sufficiently large N, Υ k (f) c k,δ/3 2 where c k,δ/3 > 0 is the constant appearing in Theorem 2.2. Proof. Let 0 < η < 1 be some parameter to be chosen later, and let f = g + h be the decomposition given by Theorem 4.1. Hence we have 0 g 2 and h is η-uniform. We would like to apply Szemerédi s theorem to the function g; however, it is bounded above by 2 rather than 1, and its density is bounded below by a function of η, which we need to be independent of our constant to be able to later take it sufficiently small. Hence we instead consider the function (g + η)/(2 + η). We now have E ( g + η 2 + η ) = E(f) E(h) + η 2 + η δ 2 + η > δ 3, since E(h) E( h ) = h U 1 h U k 1 η. Furthermore, we have 0 g + η 2 + η 1. Hence for the function (g + η)/(2 + η), the conditions of Szemerédi s theorem, Theorem 2.2, are met and we may apply it to obtain the lower bound, for N sufficiently large (depending only on k and δ) ( ) g + η Υ k (g + η) Υ k c k,δ/3 2 + η for some constant c dependent only on k and δ. Since η < 2, putting our upper bounds into the definition of Υ k gives us Υ k (f 0,..., f k 1 ) 2 k 1 η = O k (η) whenever f j = η or g for 0 j k 1, and at least one f i is equal to η. Hence Υ k (g) c k,δ O k (η). On the other hand, h U k 1 η, and g, h f +g ν+2. Hence by applying Corollary 3.1 (and Remark 3.1) we see that Υ k (f) = Υ k (g) + O k (η) + o k (1).

26 26 In particular, Υ k (f) c k,δ/3 O k (η) o k (1). By taking η small enough (depending on k and δ) we can ensure that the O k (η) term is less than c/4, and by taking N sufficiently large, we can also ensure that the o k (1) term is less than c/4. Hence, for N sufficiently large, as required. Υ k (f) c k,δ/3 2 Weak Pseudorandomness This section outlines a slightly different approach, the existence of which is hinted at by a remark in [10]. If we can assume that δ is dependent on k, then we can prove a Relative Szemerédi theorem from conditions which are strictly weaker than the pseudorandomness conditions we have been using so far. This will be important in our application to the primes, when we shall only be able to prove these weaker conditions. Let ε k be some sufficiently small constant depending only on k. Then we define weak linear pseudorandomness and weak simple pseudorandomness by changing the asymptotics and upper bounds required by adding a O(ε k ) factor. This affects our argument as follows. The Generalised von Neumann theorem is altered by a factor of O k (ε k ), using an almost identical proof. The Decomposition theorem requires the condition that η is sufficiently small depending on ε k. This is because a factor of ν 1 U k 1 is no longer o(1), but is instead O(ε k ) + o(1). By using these modified theorems in the proof of our Relative Szemerédi theorem above, we arrive at a lower bound of the form Υ k (f) c k,δ O k (η) O k (ε k ) + o(1). If δ is dependent on k, then as long as we take ε k sufficiently small, we may still arrive at the required lower bound. This gives us the following alternative Relative Szemerédi theorem, which we shall use in our application to the primes. Theorem 4.5 (Alternative Relative Szemerédi Theorem). Let k 1 and let ν be a weak k-pseudorandom function. Suppose f : Z N R satisfies 0 f(x) ν(x) for all x Z N and Ef 1 (say). Then for all sufficiently large N, 10k Υ k (f) c k 2 where c k = c k,1/30k > 0 is the constant appearing in Theorem 2.2.