Decompositions, approximate structure, transference, and the Hahn-Banach theorem

Introduction

This paper has several purposes. One is to provide a survey of some of the major recent developments in the rapidly growing field that has come to be known as additive combinatorics, focusing on three classes of theorems: decomposition theorems, approximate structural theorems and transference principles. (An explanation of these phrases will be given in just a moment.) A second is to show how the Hahn-Banach theorem leads to a simple and flexible method for proving results of these three kinds. A third is to demonstrate this by actually giving simpler proofs of several important results, or parts of results. One of the proofs we shall simplify is the proof of Green and Tao that the primes contain arbitrarily long arithmetic progressions [GT1], which leads to the fourth purpose of this paper: to provide a partial guide to their paper. We shall give a simple proof of a result that is implicit in their paper, and made explicit in a later paper of Tao and Ziegler, and then we shall explain informally how they use this result to prove their famous theorem. A proof along similar lines has been discovered independently by Reingold, Trevisan, Tulsiani and Vadhan [RTTV]. We have tried to design this paper so that the reader who is just interested in the Green-Tao theorem can get away with reading only a small part of it. However, the earlier sections of the paper provide considerable motivation for the later arguments, so such a reader would be well-advised at least to skim the sections that are not strictly speaking necessary.

Now let us describe the classes of theorems that will principally concern us. By a decomposition theorem we mean a statement that tells us that a function $f$ with certain properties can be decomposed as a sum $\sum_{i=1}^{k}g_{i}$, where the functions $g_{i}$ have certain other properties. There are two kinds of decomposition theorem that have been particularly useful. One kind says that $f$ can be written as $\sum_{i=1}^{k}g_{i}+h$, where the functions $g_{i}$ have some explicit description and $h$, the “error term” is in a useful sense small.

Another kind, which is closely related, brings us to our second class of results. An approximate structure theorem is a result that says that, under appropriate conditions, we can write a function $f$ as $f_{1}+f_{2}$, where $f_{1}$ is “structured” in some sense, and $f_{2}$ is “quasirandom”. The rough idea is that the structure of $f_{1}$ is strong enough for us to be able to analyse it reasonably explicitly, and the quasirandomness of $f_{2}$ is strong enough for many properties of $f_{1}$ to be unaffected if we “perturb” it to $f_{1}+f_{2}$. Often, in order to obtain stronger statements about the structure and the quasirandomness, one allows also a small $L_{2}$-error: that is, one writes $f$ as $f_{1}+f_{2}+f_{3}$ with $f_{1}$ structured, $f_{2}$ quasirandom, and $f_{3}$ small in $L_{2}$.

A transference principle is a statement to the effect that a function $f$ that belongs to some space $X$ of functions can be approximated by a function $g$ that belongs to another space $Y$. Such a statement is useful if the functions in $Y$ are easier to handle than the functions in $X$ and the approximation is of a kind that preserves the properties that one is interested in. As we shall see later, a transference principle is the fundamental step in the proof of Green and Tao. For now, let us merely note that a transference principle is a particular kind of decomposition theorem: it tells us that $f$ can be written as $g+h$, where $g\in Y$ and $h$ is small in an appropriate sense.

As well as the Green-Tao theorem, we shall discuss several other results in additive combinatorics. One is a structure theorem proved by Tao in an important paper [T1] that gives a discretization of Furstenberg’s ergodic-theory proof [Fu] of Szemerédi’s theorem [S1], or more precisely a somewhat different ergodic-theory argument due to Host and Kra [HK05]. We shall give an alternative proof of (a slight generalization of) this theorem, and give some idea of how it can be used to prove other results. Amongst these other results are Roth’s theorem [Rot], which states that every set of integers of positive upper density contains an arithmetic progression of length 3, and Szemerédi’s regularity lemma [S2], a cornerstone of extremal graph theory, which shows that every graph can be approximated by a disjoint union of boundedly many quasirandom graphs (and which is a very good example of an approximate structure theorem).

The remaining sections of this paper are organized as follows. The next section introduces several norms that are used to define quasirandomness. Strictly speaking, it is independent of much of the rest of the paper, since many of our results will be rather general ones about norms that satisfy various hypotheses. However, for the reader who is unfamiliar with the basic concepts of additive combinatorics it may not be obvious that these hypotheses are satisfied except in one or two very special cases: section 2 should convince such a reader that the general results can be applied in many interesting contexts.

In section 3, we introduce our main tool, the finite-dimensional Hahn-Banach theorem, and we give one or two very easy consequences of it. Even these consequences are of interest, as we shall explain—one of them is a non-trivial decomposition theorem of the first kind discussed above—but the method comes into its own when we introduce one or two further ideas in order to obtain conclusions that can be applied much more widely.

One of these ideas is the relatively standard one of polynomial approximations. Often we start with a function $f$ that takes values in an interval $[a,b]$, and we want its structured part $f_{1}$ to take values in $[a,b]$ as well. If $f_{1}$ is bounded, and if the class of structured functions is closed under composition with polynomials, then we can sometimes achieve this by choosing a polynomial $P$ such that $P(x)$ approximates $a$ when $x<a$, $x$ when $a\leq x\leq b$, and $b$ when $x>b$. Then the function $Pf_{1}$ takes values in $[a,b]$ (approximately), and under appropriate circumstances it is possible to argue that it approximates $f_{1}$. In section 4, we shall illustrate this technique by proving two results. The first is a fairly simple transference principle that we shall need later, and the second is a slightly more complicated version of it that is needed for proving the Green-Tao theorem. The latter is essentially the same as the “abstract structure theorem” of Tao and Ziegler [TZ], so called because it is an abstraction of arguments from the paper of Green and Tao. It is this second result that can be regarded as a major step in the proof of the Green-Tao theorem, and which is used to prove their transference principle. We shall end Section 4 with a brief description of the rest of the proof of Green and Tao.

In section 5, we shall prove the structure theorem of Tao mentioned earlier, and show how it leads to a strengthened decomposition theorem. We end the section, and the paper, with an indication of how to use the structure theorem.

Some basic concepts of additive combinatorics.

The Fourier inversion formula (which it is an easy exercise to verify) then tells us that

which gives the expansion of $f$ as a linear combination of characters.

and the convolution identity states that $(f*g)^{\wedge}(\psi)=\hat{f}(\psi)\hat{g}(\psi)$ for every $\psi\in\hat{G}$. That is, the Fourier transform “converts convolution into pointwise multiplication”. It also does the reverse: the Fourier transform of the pointwise product $fg$ is the convolution $\hat{f}*\hat{g}$, where the latter is defined by the formula

Parseval’s identity is the simple statement that $\langle f,g\rangle=\langle\hat{f},\hat{g}\rangle$. It is important to keep in mind that the two inner products are defined differently, one with expectations and the other with sums, just as the convolutions were defined differently in $G$ and $\hat{G}$. Setting $f=g$ in Parseval’s identity, we deduce that $\|f\|_{2}=\|\hat{f}\|_{2}$.

2. What is additive combinatorics about?

The central objects of study in additive combinatorics are finite subsets of Abelian groups. For example, one of the main results in the area, Szemerédi’s theorem, can be formulated as follows.

What are the maps of interest between finite subsets of Abelian groups? An initial guess might be that they were restrictions of group homomorphisms, but that turns out to be far too narrow a definition. Instead, they are functions called Freiman homomorphisms. A Freiman homomorphism of order $k$ between sets $A$ and $B$ is a function $\phi:A\rightarrow B$ such that

In particular, a Freiman homomorphism of order 2, often just known as a Freiman homomorphism, is a function such that $\phi(a_{1})+\phi(a_{2})=\phi(a_{3})+\phi(a_{4})$ whenever $a_{1}+a_{2}=a_{3}+a_{4}$. This is equivalent to the same definition with minus instead of plus, which is often more convenient.

A Freiman isomorphism of order $k$ is a Freiman homomorphism of order $k$ with an inverse that is also a Freiman homomorphism of order $k$. The rough idea is that a Freiman homomorphism of order $k$ preserves all the linear structure of a set $A$ that can be detected by integer combinations with coefficients adding up to 0 and with absolute values adding up to at most $2k$. For example, it is an easy exercise to show that if $A$ is an arithmetic progression and $B$ is Freiman-isomorphic to $A$, then $B$ is also an arithmetic progression. This is because a sequence $(x_{1},x_{2},\dots,x_{m})$ is an arithmetic progression, written out in a sensible order, if and only if $x_{i+2}-x_{i+1}=x_{i+1}-x_{i}$ for every $i$. It is also easy to show that if $A$ and $B$ are isomorphic, then their sumsets $A+A$ and $B+B$ have the same size.

Thus, a more precise description of the main objects studied by additive combinatorics would be that they are finite subsets of Abelian groups, up to Freiman isomorphisms of various orders.

An important aspect of results such as Szemerédi’s theorem is that they have a certain “robustness”. For instance, combining Szemerédi’s theorem with a simple averaging argument, one can deduce the following corollary (which was first noted by Varnavides [V]).

Thus, it is tempting to revise further our rough definition of additive combinatorics and say that the central objects of study are subsets of Abelian groups, up to Freiman isomorphism and “quasirandom perturbation”. However, it takes some effort to make this idea precise, since the notion of a Freiman homomorphism does not apply as well to functions as it does to sets (because it is insufficiently robust). Also, not every quantity of importance in the area is approximately invariant up to quasirandom perturbations: an example of one that isn’t is the size of the sumset $A+A$ of a set $A$ of size $n$. So we shall content ourselves with the observation that all the results of this paper are approximately invariant.

So that we can say what this means, let us give some examples of norms that measure quasirandomness.

3. Uniformity norms for subsets of finite Abelian groups.

In many respects, a quasirandom set behaves as one would expect a random set to behave, but in by no means all. For example, even if $A$ is as quasirandom as it is possible for a set to be, it does not follow that

turns out to be at least $c\delta^{3}$ for some absolute constant $c>0$. We will not prove this here (a proof can be found in [G3]), but we give the example in order to draw attention to its quadratic nature. It turns out that this feature of the example is necessary, though quite what that means is not obvious, and the proof is even less so. See subsection 2.6 for further discussion of this.

This example shows that the smallness of the $U^{2}$-norm is not sufficient to explain all the typical behaviour of a random function. For this one needs to introduce “higher” uniformity norms, of which the next one is (unsurprisingly) the $U^{3}$-norm. If $g$ is a function, then $\|g\|_{U^{3}}^{8}$ is given by the expression

From this it is easy to guess the definition of the $U^{k}$ norm, but for completeness here is a formula for it:

where $C$ denotes the operation of complex conjugation and $|\epsilon|$ denotes the number of non-zero coordinates of $\epsilon$.

Let us call a set uniform of degree $k-1$ if its $U^{k}$-norm is small. Then the above assertion is that a set of density $\delta$ that is sufficiently uniform of degree $k-1$ contains roughly as many arithmetic progressions (mod $N$) of length $k+1$ as a random set of density $\delta$ will (with high probability) contain. In particular, if $A$ is quadratically uniform (meaning that the $U^{3}$-norm of $A-\delta$ is sufficiently small), then

The arithmetic progression $\{x,x+d,\dots,x+(k-1)d\}$ can be thought of as a collection of $k$ linear forms in $x$ and $d$. It can be shown that for any collection of linear forms in any number of variables, there exists a $k$ such that every set $A$ that is sufficiently uniform of degree $k$ contains about as many of the corresponding linear configurations as a random set of the same density. This was shown by Green and Tao [GT3], who generalized the argument in [G1]. The question of precisely which $U^{k}$ norm is needed is a surprisingly subtle one. It is conjectured in [GW1] that the answer is the smallest $k$ for which the $k$th powers of the linear forms in question are linearly independent. For instance, the configuration $\{x,x+d,x+2d,x+3d\}$ needs the $U^{3}$-norm because $x-2(x+d)+(x+2d)=x^{2}-3(x+d)^{2}+3(x+2d)^{2}-(x+3d)^{2}=0$, but the cubes are linearly independent. A special case of this result is proved in [GW1] using “quadratic Fourier analysis”, which we will discuss in 2.7: to prove the full conjecture would require a theory of higher-degree Fourier analysis that will probably exist in due course but which has not yet been sufficiently developed.

4. Uniformity norms for graphs and hypergraphs.

There are very close and important parallels between uniformity of subsets of finite Abelian groups, and quasirandomness of graphs and hypergraphs. For this reason, even though the relevant parts of graph and hypergraph theory belong to extremal combinatorics, they have become part of additive combinatorics as well: one could call them additive combinatorics without the addition.

The quadruples $(x+y,x+y^{\prime},x^{\prime}+y,x^{\prime}+y^{\prime})$ are uniformly distributed over all quadruples $(a,b,c,d)$ such that $a+d=b+c$. Since the same is true of all quadruples of the form $(x,x+a,x+b,x+a+b)$, we see that the right-hand side of the above formula is nothing other than $\|f\|_{U^{2}}$.

A similar argument can be used to relate the higher-degree uniformity norms to notions of quasirandomness for $k$-uniform hypergraphs, which are like graphs except that instead of having edges, which are pairs of vertices, one has hyperedges, which are $k$-tuples of vertices. The following formula defines a norm on $k$-variable functions:

If $f(x_{1},\dots,x_{k})$ has the form $g(x_{1}+\dots+x_{k})$, then $\|f\|_{HU^{k}}=\|g\|_{U^{k}}$.

A hypergraph $H$ of density $\delta$ behaves in many respects like a random hypergraph of density $\delta$ if $\|H-\delta\|_{HU^{k}}$ is small enough. For instance, if $k=3$, then the simplex density, which is given by the expression

is roughly $\delta^{4}$, or what it would be in the random case. More generally, if $\|H_{i}-\delta\|_{HU^{k}}$ is small for $i=1,2,3,4$, then

is again roughly $\delta^{4}$. This assertion, which is proved by repeated use of the Cauchy-Schwarz inequality (see [G2] for a more general result), implies that

These ideas can be developed to give a complete proof of Szemerédi’s theorem: see [NRS], [RS], [G2], [T2].

5. Easy structure theorems for the U2U^{2}-norm.

A great deal of information about the $U^{2}$-norm comes from the following simple observation.

By the convolution identity and Parseval’s identity,

The result follows on taking fourth roots. ∎

Now let us suppose that $\|f\|_{2}\leq 1$, and let us fix some small constant $\eta>0$. Then the number of characters $\psi$ such that $|\hat{f}(\psi)|\geq\eta$ is at most $\eta^{-2}$, since $\sum_{\psi}|\hat{f}(\psi)|^{2}=\|\hat{f}\|_{2}^{2}=\|f\|_{2}^{2}=1$. Using the inversion formula and this fact, we can split $f$ into two parts, $\sum_{\psi\in K}\hat{f}(\psi)\psi$ and $\sum_{\psi\notin K}\hat{f}(\psi)\psi$, where $K$ is the set of all $\psi$ such that $|\hat{f}(\psi)|\geq\eta$. Let us call these two parts $g$ and $h$, respectively. The function $g$ involves a bounded number of characters, and characters are functions that we can describe completely explicitly. Therefore, it can be thought of as the “structured” part of $f$. As for $h$, it is quasirandom, since

Unfortunately, this simple decomposition turns out not to be very useful, for reasons that we shall explain later. We mention it in order to put some of our later results in perspective. The same applies to the next result, which shows that one can obtain a much stronger relationship between $\|h\|_{U^{2}}$ and the upper bound on the size of $K$ if one is prepared to tolerate a small $L_{2}$-error as well. This result and its proof are part of the standard folklore of additive combinatorics.

Let $N=|G|$ and let us enumerate the dual group $\hat{G}$ as $\psi_{1},\dots,\psi_{N}$ in such a way that the absolute values of the Fourier coefficients $\hat{f}(\psi_{i})$ are in non-increasing order. Choose an increasing sequence of positive integers $m_{1},m_{2},\dots$ in such a way that $m_{r+1}\geq\eta(m_{r})^{-4}$ for every $r$.

Now let us choose $i$ and attempt to prove the result using the decomposition $f_{1}=\sum_{i\leq m_{r}}\hat{f}(\psi_{i})\psi_{i}$, $f_{2}=\sum_{i>m_{r+1}}\hat{f}(\psi_{i})\psi_{i}$, and $f_{3}=\sum_{m_{r}<i\leq m_{r+1}}\hat{f}(\psi_{i})\psi_{i}$. Then $f_{1}$ is a linear combination of at most $m_{r}$ characters. Since there can be at most $m_{r+1}$ characters $\psi$ with $|\hat{f}(\psi)|\geq m_{r+1}^{-1/2}$, we find that

Therefore, we are done if $\|f_{3}\|_{2}\leq\epsilon$. But the possible functions $f_{3}$ (as $r$ varies) are disjoint parts of the Fourier expansion of $f$, so at most $\epsilon^{-2}$ of them can have norm greater than $\epsilon$. Therefore, we can find $r\leq\epsilon^{-2}$ such that the proposed decomposition works. ∎

There is nothing to stop us taking $m_{1}=1$. The proof then gives us the desired decomposition for some $m$ that is bounded above by a number that results from starting with 1 and applying the function $t\mapsto\eta(t)^{-4}$ at most $\epsilon^{-2}$ times. Although Proposition 2.5 is still not all that useful, it resembles other results that are, as we shall see in due course. In those results, it is common to require $\eta(m)$ to be exponentially small: the resulting bound is then of tower type.

6. Inverse theorems.

A remarkable inverse theorem, which lies at the heart of many recent results in additive combinatorics, is a theorem of Freiman [F] (later given a considerably more transparent proof by Ruzsa [Ru]) that characterizes sets that have small sumsets. If $A$ is a set of $n$ integers, then it is easy to show that the sumset $A+A$ has size at least $2n-1$ and at most $n(n+1)/2$. What can be said about $A$ if the size of the sumset is close to its minimum, in the sense that $|A+A|\leq C|A|$ for some fixed constant $C$? A simple example of such a set is an arithmetic progression. A slightly less simple example is a set $A$ that is contained in an arithmetic progression of length at most $Cn/2$. A less simple example altogether is a “two-dimensional arithmetic progression”: that is, a set of the form $\{x_{0}+rd_{1}+sd_{2}:0\leq r<t_{1},0\leq s<t_{2}\}$. If $A$ is such a set, then $|A+A|\leq 4|A|$, and more generally if $A$ is a $k$-dimensional arithmetic progression (the definition of which is easy to guess), then $|A+A|\leq 2^{k}|A|$. As in the one-dimensional case, one can pass to large subsets and obtain more examples. Freiman’s theorem states that one has then exhausted all examples.

For every $C$ there exist $k$ and $K$ such that every set $A$ of $n$ integers such that the sumset $A+A$ has size at most $Cn$ is contained in an arithmetic progression of dimension at most $k$ and cardinality at most $Kn$.

Freiman’s theorem has been extremely influential, in large part because of Ruzsa’s proof, which was extremely elegant and conceptual, and gave much better bounds than Freiman’s argument. These bounds have subsequently been improved by Chang [C], who added further interesting ingredients to Ruzsa’s argument. A generalization of Freiman’s theorem to subsets of an arbitrary Abelian group was proved by Green and Ruzsa [GR].

The notion of an inverse theorem makes sense also for functions defined on Abelian groups. For instance, here is a simple inverse theorem about functions with large $U^{2}$-norm.

By Lemma 2.4 and our assumptions about $f$,

Conversely, and without any assumption about $\|f\|_{2}$, if there exists a character $\psi$ such that $\langle f,\psi\rangle\geq c$, then $\|f\|_{U^{2}}=\|f\|_{4}\geq c^{1/4}$. Therefore, correlation with a character “explains” the largeness of the $U^{2}$-norm.

implies easily that $\|f\|_{U^{3}}=1$. However, it is also easy to show that $f$ does not correlate significantly with any character. Therefore, we are forced to consider quadratic functions. If $q$ is a quadratic function, then let us call the function $\omega^{q}$ a quadratic phase function.

In [G1], the following “weak inverse theorem” was proved for all $U^{k}$ norms.

This result was the main step in the proof of Szemerédi’s theorem given in [G1]. The reason that this is a “weak inverse theorem” is that the converse is far from true. The result shows that $f$ correlates with a function that is made out of many fragments of polynomial phase functions, but it does not provide what one might hope for: correlation with a single generalized polynomial phase function. However, the proof strongly suggested that such a result should be true, and Green and Tao, by adding some important further ingredients, have established a strong inverse theorem in the quadratic case [GT2]. Let us state their result a little imprecisely.

7. Higher Fourier analysis.

However, one can still hope for a structure theorem that resembles Proposition 2.5. We would expect it to say that a function $f$ can be decomposed into a linear combination of a small number of generalized quadratic phase functions, plus a function with very small $U^{3}$-norm, plus a function that is small in $L_{2}$. Green and Tao deduced such a result from their inverse theorem, and thereby initiated a form of quadratic Fourier analysis. In [GW1], a different method was given for deducing somewhat different decomposition theorems from inverse theorems. The main ingredient of this method was the Hahn-Banach theorem: the proof will be sketched in the next section. This gave an alternative form of quadratic Fourier analysis, which provided much better bounds for the results of that paper (the ones that concerned controlling systems of linear forms with $U^{k}$-norms).

We shall have more to say about higher Fourier analysis later in the paper.

8. Easy structure theorems for graphs.

We have already seen that the $U^{2}$-norm of the one-variable function $g$, defined on a finite Abelian group $G$, can be regarded as the $GU^{2}$-norm of the two-variable function $f(x,y)=g(x+y)$. The relationship does not stop here, however. If $\psi$ is a character, then, for any $x$,

This shows that characters are similar to eigenvectors of the symmetric matrix $f(x,y)$, except that they are mapped to multiples of their complex conjugates. However, it is notable that the corresponding “eigenvalues” are the Fourier coefficients of the function $g$. This observation suggests, correctly as it turns out, that eigenvalues play a similar role for real symmetric matrices to the role played by Fourier coefficients for functions defined on finite Abelian groups.

The finite-dimensional spectral theorem tells us that a real symmetric matrix $f(x,y)$ has an orthonormal basis of eigenvectors. If these are $u_{1},\dots,u_{n}$ and the corresponding eigenvalues are $\lambda_{1},\dots,\lambda_{n}$, then we can express this by saying that

by the orthonormality (with respect to the $L_{2}$-norm) of the eigenvectors.

Let $X$ be a finite set and let $f$ be a symmetric real-valued function defined on $X^{2}$. Let the eigenvalues of $f$ be $\lambda_{1},\dots,\lambda_{n}$. Then $\|f\|_{GU^{2}}^{4}=\sum_{r}\lambda_{r}^{4}$.

All results of this kind are proved by expanding the expression for $\|f\|_{GU^{2}}^{4}$ in terms of the spectral decomposition $\sum_{r}\lambda_{r}u_{r}\otimes u_{r}$ of $f$.

A similar but easier proof establishes that $\|f\|_{2}^{2}=\sum_{r}\lambda_{r}^{2}$.

The next result is a direct analogue for symmetric two-variable real functions (and therefore in particular for graphs) of Proposition 2.5

Let $N=|X|$ and let us enumerate an orthonormal basis $(u_{i})_{1}^{N}$ of eigenvectors of $f$ in such a way that the absolute values of the eigenvalues $\lambda_{i}$ are in non-increasing order. Choose an increasing sequence of positive integers $m_{1},m_{2},\dots$ in such a way that $m_{r+1}\geq\eta(m_{r})^{-4}$ for every $r$.

Now let us choose $i$ and attempt to prove the result using the decomposition $f_{1}=\sum_{i\leq m_{r}}\lambda_{i}u_{i}\otimes u_{i}$, $f_{2}=\sum_{i>m_{r+1}}\lambda_{i}u_{i}\otimes u_{i}$, and $f_{3}=\sum_{m_{r}<i\leq m_{r+1}}\lambda_{i}u_{i}\otimes u_{i}$. Then $f_{1}$ is a linear combination of at most $m_{r}$ eigenvectors, which are orthonormal to each other by the spectral theorem. By the remark above about the sum of the squares of the eigenvalues, there can be at most $m_{r+1}$ eigenvectors $u_{i}$ with $|\lambda_{i}|\geq m_{r+1}^{-1/2}$. Therefore,

Therefore, we are done if $\|f_{3}\|_{2}\leq\epsilon$. But the possible functions $f_{3}$ (as $r$ varies) are disjoint parts of the spectral expansion of $f$, so at most $\epsilon^{-2}$ of them can have norm greater than $\epsilon$. Therefore, we can find $r\leq\epsilon^{-2}$ such that the proposed decomposition works. ∎

The above result is closely related to a “weak regularity lemma” due to Frieze and Kannan [FK].

The Hahn-Banach theorem and simple applications.

Let us begin by stating the version of the Hahn-Banach theorem that we shall need.

Now let us prove two corollaries, both of which are useful for proving decomposition theorems.

Let $K$ be the convex body $\sum_{i}c_{i}K_{i}$. Our hypothesis is that $f\notin K$. Since $K$ is closed, it follows that there exists $\epsilon>0$ such that $(1+\epsilon)^{-1}f\notin K$. Therefore, by Theorem 3.1, there is a constant $\beta$ and a linear functional $\phi$ such that $(1+\epsilon)^{-1}\langle f,\phi\rangle\geq\beta$ and $\langle g,\phi\rangle\leq\beta$ for every $g\in K$. Again using the fact that $K$ is closed, we can add a small Euclidean ball $B$ to $K$ in such a way that $(1+\epsilon)^{-1}f\notin B+K$. Since $0\in K$, it follows that $\beta>0$. Therefore, we can divide $\phi$ by $\beta$ and get $\beta$ to be 1, with the result that $\langle f,\phi\rangle\geq(1+\epsilon)\beta$. Since $0$ belongs to each $K_{i}$, we can also conclude that $\langle g,\phi\rangle\leq 1$ for every $g\in c_{i}K_{i}$, which completes the proof. ∎

Let $K$ be the set of all convex combinations $c_{1}f_{1}+\dots+c_{r}f_{r}$ with $f_{i}\in K_{i}$. Then $K$ is a closed convex set and $f$ is not contained in $K$. Therefore, there exists $\epsilon>0$ such that $(1+\epsilon)^{-1}f\notin K$. By Theorem 3.1 there is a functional $\phi$ and a constant $\beta$ such that $(1+\epsilon)^{-1}\langle f,\phi\rangle\geq\beta$ and $\langle g,\phi\rangle\leq\beta$ whenever $g$ belongs to $K$. In particular, $\langle g,\phi\rangle\leq\beta$ whenever $g$ belongs to one of the sets $K_{i}$. As in the proof of the previous corollary, $\beta$ must be positive and can therefore be assumed to be 1. The result follows. ∎

Then this formula does indeed define a norm, and its dual norm $\|.\|^{*}$ is given by the formula

It is a simple exercise to check that the expression does indeed define a norm.

Let us begin by supposing that $\|z\|_{\sigma}^{*}\geq 1$ for some $\sigma\in\Sigma$. Then there exists $x\in V_{\sigma}$ such that $\|x\|_{\sigma}\leq 1$ and $|\langle x,z\rangle|\geq 1$. But then $\|x\|\leq 1$ as well, from which it follows that $\|z\|^{*}\geq 1$. Therefore, $\|z\|^{*}$ is at least the maximum of the $\|z\|_{\sigma}^{*}$.

Now let us suppose that $\|z\|^{*}>1$. This means that there exists $x$ such that $\|x\|\leq 1$ and $|\langle x,z\rangle|\geq 1+\epsilon$ for some $\epsilon>0$. Let us choose $x_{1},\dots,x_{k}$ such that $x_{i}\in V_{\sigma_{i}}$ for each $i$, $x_{1}+\dots+x_{k}=x$, and $\|x_{1}\|_{\sigma_{1}}+\dots+\|x_{k}\|_{\sigma_{k}}<1+\epsilon$. Then

so there must exist $i$ such that $|\langle x_{i},z\rangle|>\|x_{i}\|_{\sigma_{i}}$, from which it follows that $\|z\|_{i}^{*}>1$. This proves that $\|z\|^{*}$ is at most the maximum of the $\|z\|_{i}^{*}$. ∎

Then this formula does indeed define a norm, and its dual norm $\|.\|^{*}$ is defined by the formula $\|f\|^{*}=max\{|\langle f,\sigma\rangle|:\sigma\in\Sigma\}$.

We now prove a very simple (and known) decomposition result that illustrates our basic method.

Suppose that the result is false. We shall apply Corollary 3.3 to the function $f/\|f\|_{2}$, with $K_{1}$ and $K_{2}$ taken to be the unit balls of $\|.\|$ and $\|.\|^{*}$. Our hypothesis is equivalent to the assertion that $f/\|f\|_{2}$ is not a convex combination $c_{1}g_{1}+c_{2}g_{2}$ with $g_{i}\in K_{i}$ for $i=1,2$. Therefore, we obtain a functional $\phi$ such that $\langle f,\phi\rangle>\|f\|_{2}$ and $\|\phi\|^{*}$ and $\|\phi\|$ are both at most 1. But the first property implies, by the Cauchy-Schwarz inequality, that $\|\phi\|_{2}>1$, while the second implies that $\|\phi\|_{2}^{2}=\langle\phi,\phi\rangle\leq\|\phi\|\|\phi\|^{*}\leq 1$. This is a contradiction. ∎

A simple modification of Proposition 3.6 makes it a little more flexible. Suppose that we wish to write $f$ as $g+h$ with $\|g\|$ small and $\|h\|^{*}$ not too large. If we define a new norm $|.|$ to be $\epsilon^{-1}\|.\|$, then $|.|^{*}=\epsilon\|.\|^{*}$. Applying Proposition 3.6 to these rescaled norms, we find that we can write $f$ as $g+h$ in such a way that $\epsilon^{-1}\|g\|+\epsilon\|h\|^{*}\leq\|f\|_{2}$. In particular, if $\|f\|_{2}=1$, then $\|g\|\leq\epsilon$ and $\|h\|^{*}\leq\epsilon^{-1}$.

The reason such a result might be expected to be useful in additive combinatorics is that, as demonstrated in the previous section, we have a good supply of norms $\|.\|$ that measure quasirandomness. Moreover, their duals, as we shall see later, can be thought of as a sort of measure of structure. Perhaps the simplest example that illustrates this is if we look at functions $f$ defined on finite Abelian groups, and take $\|f\|$ to be $\|\hat{f}\|_{\infty}$. If $\|f\|_{2}\leq 1$, then

a calculation we have already done. This shows that for functions with bounded $L_{2}$-norm there is a rough equivalence between $\|f\|_{U^{2}}$ and $\|\hat{f}\|_{\infty}$, in the sense that if one is small then so is the other.

Thus, if $\|f\|_{2}\leq 1$ then $\|f\|=\|\hat{f}\|_{\infty}$ being small tells us that $f$ is quasirandom. The dual norm, $\|f\|^{*}=\|\hat{f}\|_{1}$, is a sort of measure of structure, since if $\|\hat{f}\|_{1}$ is at most $C$, then $f$ is a small multiple of a convex combination of trigonometric functions, which can be approximated in $L_{2}$ by a linear combination of a bounded number of such functions. Thus, we recover a result that resembles Proposition 2.5. It is weaker, however, because we have not yet related the quasirandomness constant to the structure constant by means of an arbitrary function. However, this is easily done, again with the help of an $L_{2}$ error term, as the next result shows.

In particular, $\|f_{1}\|^{*}\leq C_{i}$, $\|f_{2}\|\leq\eta(C_{i})$ and $\|f_{3}\|_{2}\leq\epsilon$.

If there is no such decomposition for $i$, then by Corollary 3.3 there is a functional $\phi_{i}$ such that $\|\phi_{i}\|\leq C_{i}^{-1}$, $\|\phi_{i}\|^{*}\leq\eta(C_{i})^{-1}$, $\|\phi_{i}\|_{2}\leq\epsilon^{-1}$, and $\langle\phi_{i},f\rangle>1$. If this is true for every $i\leq r$ then

where the first inequality follows from Cauchy-Schwarz and the assumption that $\|f\|_{2}\leq 1$.

the last inequality following from the way we constructed the sequence $C_{1},\dots,C_{r}$. Therefore,

This contradicts the previous estimate, since $r\geq 2\epsilon^{-1}$. ∎

It is easy to deduce Proposition 2.5 from this result. Of course, this fact on its own is not a very convincing demonstration of the utility of the Hahn-Banach theorem, since for the norm $\|f\|=\|\hat{f}\|_{\infty}$ it is easy to write down an explicit decomposition of $f$, as we saw in section 2.5. But there are other important norms where this is certainly not the case. For example, if we take $\|f\|$ to be $\|f\|_{U^{k}}$ for a larger $k$, then there is no obvious decomposition of $f$ from which we can read off three functions $f_{1}$, $f_{2}$ and $f_{3}$ with the required properties.

2. Deducing decomposition theorems from inverse theorems.

The main result of [GW1] shows that certain linear configurations occur with the “expected” frequency in any set $A$ for which the balanced function $A-\delta$ (where $\delta$ is the density of $A$) has sufficiently small $U^{2}$-norm. The interest in the result is that the $U^{2}$-norm suffices for the configurations in question, whereas the natural arguments that generalize the proof that the $U^{k}$ norm controls progressions of length $k-2$ would suggest that the $U^{3}$-norm was needed. In order to prove the result, a form of quadratic Fourier analysis was needed, as we have already mentioned. The approach in [GW1] was to apply directly a result of Green and Tao, which obtains a decomposition of a bounded function $f$ by constructing an averaging projection $P$ with the property that $f-Pf$ has small $U^{3}$ norm. However, there was a technicality involved that forced us to use an iterated version of their result that gives rise to very weak bounds. In order to obtain reasonable bounds for the problem, it turned out to be convenient—indeed, as far as we could tell, necessary—to prove a decomposition theorem that could be regarded as a quadratic analogue of Proposition 2.5, with the important difference that the strong dependence of $\eta(m)$ on $m$ was not needed. (This was why it was possible to obtain good bounds.)

The argument appears in [GW2], and it can be regarded as a special case of a general principle that can be informally summarized as follows: to each inverse theorem there is a corresponding decomposition theorem. It is possible to give a formal statement, as will be clear from our discussion, but in practice it is much easier to describe a method for deducing decompositions from inverse theorems than it is to state an artificial lemma that declares that the method works. The main reason for this is that when one applies the method, one typically starts with the decomposition one wants to prove and the inverse theorem one can prove, and adjusts the former until it follows from the latter. We shall reflect this in our discussion below: more precisely, we shall assume that a decomposition of a certain general kind does not exist, draw an easy consequence from this, and see when this consequence contradicts any given inverse theorem.

The function $f$ can be written in the form

where $\sum_{i=1}^{k}|\lambda_{i}|\leq M$, each $\sigma_{i}$ belongs to $\Sigma$, and for each $j\leq r$ we have an inequality of the form $\|g_{j}\|_{(j)}\leq\eta_{j}$.

Typically, $r$ will be a very small integer such as 2.

defines a norm, and that the dual of this norm is the norm

Now let us suppose that no decomposition of the kind we are looking for exists. This is equivalent to the assumption that $f$ has no decomposition of the form $g_{0}+g_{1}+\dots+g_{k}$ with $\|g_{0}\|\leq M$ and $\|f_{i}\|_{(i)}\leq\eta_{i}$ for every $i$. If this is the case, then by Corollary 3.2 there must be a linear functional $\phi$ such that $\langle f,\phi\rangle>1$, $\|\phi\|^{*}\leq M^{-1}$, and $\|\phi\|_{(i)}^{*}\leq\eta_{i}^{-1}$ for $i=1,2,\dots,r$.

The statement that $\|\phi\|^{*}\leq M^{-1}$ tells us that $|\langle\sigma,\phi\rangle|\leq M^{-1}$ for every $\sigma\in\Sigma$. Thus, what we would like is an inverse theorem that concludes the opposite: that there must be some $\sigma\in\Sigma$ such that $|\langle\sigma,\phi\rangle|>M^{-1}$. Before we think about this, let us list the assumptions that we have at our disposal.

Suppose that there is no decomposition $f=\sum_{i=1}^{k}\lambda_{i}\sigma_{i}+g_{1}+\dots+g_{r}$ such that $\sum_{i=1}^{k}|\lambda_{i}|\leq M$, each $\sigma_{i}$ belongs to $\Sigma$, and $\|g_{j}\|_{(j)}\leq\eta_{j}$ for each $j\leq r$. Then there exists $\phi$ such that

(i) $\langle\sigma,\phi\rangle\leq M^{-1}$ for every $\sigma\in\Sigma$;

(iii) $\|\phi\|_{(j)}^{*}\leq\eta_{j}^{-1}$ for $j=1,2,\dots,r$.

The assumptions of an inverse theorem are typically that $f$ is not too big in one norm, such as, for instance, the $L_{\infty}$-norm, but not too small in another, such as the $U^{3}$-norm. The only information we have that could possibly imply a lower bound on any norm of $\phi$ is the inequality $\langle f,\phi\rangle>1$, and even that does not help unless we have an upper bound on some norm of $f$. (Of course, it is hardly surprising that such a bound would be required for a theorem that allows us to decompose $f$ into a bounded combination of bounded functions.)

So let us suppose that we have an inverse theorem of the following form. (We have introduced the constant $K$ to allow us to multiply $\phi$ by an arbitrary non-zero scalar.)

This will be contradicted under the following circumstances:

(a) the upper bounds $\|\phi\|_{(i)}^{*}\leq\eta_{i}^{-1}$ imply that $\|\phi\|\leq K$;

(b) the upper bounds on the $\|\phi\|_{(i)}^{*}$, an upper bound on some norm of $f$, and the lower bound $\langle f,\phi\rangle>1$, together imply that $|||\phi|||\geq\epsilon$;

For example, suppose that $M^{-1}<Kc(\epsilon\eta)$ and we would like a decomposition $f=\sum_{i=1}^{k}\lambda_{i}\sigma_{i}+g+h$ with $\sum_{i=1}^{k}|\lambda_{i}|\leq M$, $|||g|||\leq\epsilon$ and $\|h\|^{*}\leq\eta$. If such a decomposition does not exist, then we obtain $\phi$ such that $\langle\sigma,\phi\rangle<\eta^{-1}c(\epsilon\eta)$ for every $\sigma\in\Sigma$, $|||\phi|||^{*}\leq\epsilon^{-1}$, $\|\phi\|\leq\eta^{-1}$, and $\langle f,\phi\rangle>1$. If we also know that $\|f\|_{2}\leq 1$, then it follows that $\|\phi\|_{2}\geq 1$. But since $\|\phi\|_{2}^{2}\leq|||\phi|||.|||\phi|||^{*}$, it follows that $|||\phi|||\geq\epsilon$. This contradicts the inverse theorem (with $K=\eta^{-1}$).

If we know a little bit more about $f$, then we can obtain a correspondingly stronger result. For instance, suppose that we know that $|||f|||^{*}\leq\epsilon^{-1}$. Then the bound $\langle f,\phi\rangle>1$ immediately implies that $|||\phi|||>\epsilon$, so we do not need the error term $g$ in the decomposition.

Decomposition results obtained by the simple argument above—just assume that a decomposition doesn’t exist, apply Hahn-Banach, and contradict an inverse theorem—can be very useful. However, in order to use them one has to do a little more work. For example, it is not usually trivial that a sum of the form $\sum_{i=1}^{k}\lambda_{i}\sigma_{i}$ is “structured”, even if the sum $\sum_{i=1}^{k}|\lambda_{i}|$ is smallish and all the individual functions $\sigma_{i}$ are highly structured. The difficulty is that $k$ may be very large, and in order to deal with it one tends to need a principle that says that functions $\sigma_{i}$ are either “closely related” or “far apart”. A simple example is when $\Sigma$ is the set of all characters, in which case any two elements of $\Sigma$ are either identical or orthogonal. In [GW2] a lemma was proved to the effect that two generalized quadratic phases were either “linearly related” or “approximately orthogonal”. That made it possible to replace the linear combination by a much smaller linear combination of slightly more general functions.

A second point is that one sometimes wants more information about the “structured function” $f_{1}=\sum_{i=1}^{k}\lambda_{i}\sigma_{i}$. For instance, if $\|f\|_{\infty}\leq 1$ it can be extremely helpful to know that $\|f_{1}\|_{\infty}\leq 1$ as well. This does not come directly out of the method above, but it does when we combine that method with methods that we shall discuss in the next section.

with the following property: each $\sigma_{i}$ belongs to $\Sigma$ and there is a constant $C\leq C_{0}$ such that $\sum_{i=1}^{k}|\lambda_{i}|\leq C$, $\|f_{2}\|\leq\eta(C)$, and $\|f_{3}\|_{1}\leq\eta$.

The positivity and boundedness problems.

In section 5 we shall prove a very general result of this kind. In this section, we shall prove some simpler results that illustrate the method of polynomial approximations; we shall use this method repeatedly later.

To begin with, we need a definition that will pick out the class of norms for which we can prove results. Actually, for now we shall give a definition that is not always broad enough to be useful. In the next section we shall define a broader class of norms to which the method still applies.

We shall use the following simple lemma repeatedly.

For the second part, if $\|f\|^{*}\leq 1$ then $\|f^{n}\|^{*}\leq 1$ for every $n$. It follows that $\|f\|_{\infty}\leq 1$, since otherwise at least one coordinate of $f_{n}$ would be unbounded. Therefore, $\|f\|_{\infty}\leq\|f\|^{*}$ for every $f$. ∎

It is immediate from the definition of $\rho(C,\delta,J)$ that for every $C$ and every $\delta>0$ there exists a polynomial $P$ such that $|P(x)-J(x)|\leq\delta$ for every $x\in[-C,C]$, and such that $R_{P}(C)\leq\rho(C,\delta,J)$.

Suppose that $P$ is the polynomial $P(x)=a_{n}x^{n}+\dots+a_{1}x+a_{0}$. Then, by the triangle inequality and the algebra property of $\|.\|^{*}$,

Since the coefficients of $R_{P}$ are all non-negative, this is at most $R_{P}(C)$, which is at most $\rho(C,\delta,J)$, by our choice of $P$. ∎

In more qualitative terms, the above lemma tells us that if $\phi$ is bounded in an algebra norm and we compose it with an arbitrary continuous function $J$, then the resulting function $J\phi$ can be uniformly approximated by functions that are still bounded in the algebra norm.

The next result is our first transference theorem of the paper. It tells us that if $\mu$ and $\nu$ are non-negative functions on a set $X$ and they are sufficiently close in an appropriate norm, then any non-negative function that is dominated by $\mu$ can be “transferred to”—that is, approximated by—a non-negative function that is dominated by $\nu$. We shall apply this principle in Section 5. As we shall see later in this section, it is also not hard to generalize the result to obtain a generalized version of the Green-Tao transference theorem.

An equivalent way of stating the conclusion is that $f=g+h$ with $0\leq g\leq\nu(1-\delta)^{-1}$ and $\|h\|\leq\eta$. Thus, if the result is false then we can find a functional $\phi$ such that $\langle f,\phi\rangle>1$, but $\langle g,\phi\rangle\leq 1$ for every $g$ such that $0\leq g\leq\nu(1-\delta)^{-1}$, and $\|\phi\|^{*}\leq\eta^{-1}$.

The first condition on $\phi$ is equivalent to the statement that $\langle\nu,\phi_{+}\rangle\leq 1-\delta$. To see this, note that for any $\phi$, the $g$ that maximizes $\langle g,\phi\rangle$ takes the value $0$ when $\phi(x)<0$ and $\nu(x)(1-\delta)^{-1}$ when $\phi(x)\geq 0$, in which case $\langle g,\phi\rangle=(1-\delta)^{-1}\langle\nu,\phi_{+}\rangle$.

Now $\phi_{+}$ is equal to $J\phi$. Since $\|.\|^{*}$ is an algebra norm, we can apply Lemma 4.2 and obtain a polynomial $P$ such that $\|P\phi-\phi_{+}\|_{\infty}\leq\delta/4$ and $\|P\phi\|^{*}\leq R_{P}(C)=\rho(\eta^{-1},\delta/4,J)$, which we shall abbreviate to $\rho$.

Since $\langle\nu,\phi_{+}\rangle\leq 1-\delta$ and $\|\nu\|_{1}\leq 1$, it follows that $\langle\nu,P\phi\rangle\leq 1-3\delta/4$. Since $\|P\phi\|^{*}\leq\rho$ and $\|\mu-\nu\|\leq\epsilon$, it follows that $\langle\mu,P\phi\rangle\leq 1-3\delta/4+\epsilon\rho$. Since $\|\mu\|_{1}\leq 1$, it follows that $\langle\mu,\phi_{+}\rangle\leq 1-\delta/2+\epsilon\rho$. Since $f\leq\mu$ it follows that $\langle f,\phi_{+}\rangle\leq 1-\delta/2+\epsilon\rho$, and since $f\geq 0$ it follows that $\langle f,\phi\rangle\leq 1-\delta/2+\epsilon\rho$, which is a contradiction. ∎

2. Approximate duality and algebra-like structures.

As the previous section shows, we can carry out polynomial-approximation arguments when we are looking at a norm $\|.\|$ for which the dual norm $\|.\|^{*}$ is an algebra norm. A key insight of Green and Tao (which has received less comment than other aspects of their proof) is that one can carry out polynomial-approximation arguments under hypotheses that are weaker in two respects: one can use pairs of norms that are not precisely dual to each other, and the norm that measures structure can have much weaker properties than those of an algebra norm. It is not hard to generalize the arguments in an appropriate way: the insight was to see that there were important situations in which one could obtain the weaker hypotheses even when the stronger ones were completely false.

To see why this might be, think once again about the one algebra norm we have so far considered, namely $\|\hat{f}\|_{\infty}$. For bounded functions $f$, this is closely related (by Proposition 2.7 and the remark after it) to $\|\hat{f}\|_{4}$, which equals the $U^{2}$-norm, so we can deduce facts related to the $U^{2}$-norm from the fact that $\|\hat{f}\|_{1}$ is an algebra norm.

We can regard this argument as carrying out the following procedure. First, we establish an inverse theorem for the $U^{2}$-norm: this is what we did in Proposition 2.7. We then note that the functions that we obtain in the inverse theorem, namely the characters, are closed under pointwise multiplication. And then we make the following observation.

Suppose that $f=\sum_{i=1}^{k}\lambda_{i}f_{i}$ and $g=\sum_{j=1}^{l}\mu_{j}g_{j}$, with all $f_{i}$ and $g_{j}$ in $X$. Then $fg=\sum_{i=1}^{k}\sum_{j=1}^{l}\lambda_{i}\mu_{j}f_{i}g_{j}$. Since $X$ is closed under pointwise multiplication, each $f_{i}g_{j}$ belongs to $X$. Moreover, $\sum_{i=1}^{k}\sum_{j=1}^{l}|\lambda_{i}||\mu_{j}|=\sum_{i=1}^{k}|\lambda_{i}|\sum_{j=1}^{l}|\mu_{j}|$. From this the submultiplicativity follows easily. The fact that $\|\mathbf{1}\|=1$ follows from the assumption that $\mathbf{1}\in X$ and that all functions in $X$ have $L_{\infty}$-norm at most 1. ∎

Now suppose that we want to prove comparable facts about the $U^{3}$-norm. An obvious approach would be to use Theorem 2.9, the inverse theorem for the $U^{3}$-norm. However, the generalized quadratic phase functions that appear in the conclusion of that theorem are not quite closed under pointwise multiplication: associated with them are certain parameters that one wants to be small, which obey rules such as $\gamma(fg)\leq\gamma(f)+\gamma(g)$.

As we shall see, this is not a serious difficulty, because often one can restrict attention to products of a bounded number of functions that an inverse theorem provides. A more fundamental problem is that for the higher $U^{k}$-norms we do not (yet) have an inverse theorem. Or at least, we do not have an inverse theorem where the function that appears in the conclusion can be explicitly described. What Green and Tao did to get round this difficulty was to define a class of functions that they called basic anti-uniform functions, and to prove a “soft” inverse theorem concerning those functions.

Needless to say, the above definition generalizes straightforwardly to a class of basic anti-uniform functions for the $U^{k}$-norm, for any $k$. The same applies to the next proposition.

The way we have stated the result is rather artificial, since the basic anti-uniform function in question is nothing other than $\mathcal{D}f$. Moreover, it is trivial that $\langle f,\mathcal{D}f\rangle\geq\epsilon^{8}$, since if we expand the left-hand side we obtain the formula for $\|f\|_{U^{3}}^{8}$. ∎

Of course, the price one pays for such a simple proof is that one has far less information about basic anti-uniform functions than one would have about something like a generalized polynomial phase function. In particular, it is not obvious what one can say about products of basic anti-uniform functions.

We remark here that an inequality proved in [G1] implies easily that $\langle g,\mathcal{D}f\rangle\leq\|g\|_{U^{3}}\|f\|_{U^{3}}^{7}$ for every function $g$. Thus, $\|\mathcal{D}f\|_{U^{3}}^{*}\leq\|f\|_{U^{3}}^{7}$. Since $\langle f,\mathcal{D}f\rangle=\|f\|_{U^{3}}\|f\|_{U^{3}}^{7}$, we see that $\mathcal{D}f$ is a support functional for $f$. It is not hard to show that it is unique (up to a scalar multiple). Since every function is a support functional for something, it may seem as though there is something odd about the definition of a basic anti-uniform function. However, it is less all-encompassing than it seems, because we are restricting attention to functions $\mathcal{D}f$ for which $f$ belongs to some specified class of functions $X$. (Nevertheless, we shall usually choose $X$ in such a way that every function is a multiple of a basic anti-uniform function.)

A crucial fact that Green and Tao proved about basic anti-uniform functions is that, for suitable sets $X$, their products have $(U^{k})^{*}$-norms that can be controlled. To be precise, they proved the following lemma. (It is not stated as a lemma, but rather as the beginning step in the proof of their Lemma 6.3.)

For every positive integer $K$ there is a constant $C_{K}$ such that if $\mathcal{D}f_{1},\dots,\mathcal{D}f_{K}$ are basic anti-uniform functions [with respect to a suitable set $X$], then $\|\mathcal{D}f_{1}\dots\mathcal{D}f_{K}\|_{U^{k}}^{*}\leq C_{K}$.

3. A generalization of the Green-Tao-Ziegler transference theorem.

We shall be more concerned with the form of Lemma 4.6 than with the details of what $X$ is, since our aim is to describe in an abstract way the important properties of the operator $f\mapsto\mathcal{D}f$. This we do in the next definition, which is meant to capture the idea that the dual of a certain norm somewhat resembles an algebra norm.

(i) $\langle f,\mathcal{D}f\rangle\leq 1$ for every $f\in X$;

(ii) $\langle f,\mathcal{D}f\rangle\geq c(\epsilon)$ for every $f\in X$ with $\|f\|\geq\epsilon$;

(iii) $\|\mathcal{D}f_{1}\dots\mathcal{D}f_{K}\|^{*}\leq C(K)$ for any functions $f_{1},\dots,f_{K}\in X$.

By Lemma 3.5 the dual of the norm $\|.\|_{BAC}$ is given by the formula

Thus, it measures the ease with which a function can be decomposed into a linear combination of basic anti-uniform functions. In terms of this norm, property (ii) above is telling us that if $f\in X$ and $\|f\|\geq\epsilon$ then $\|f\|_{BAC}\geq c(\epsilon)$. This expresses a rough equivalence between the two norms, of a similar kind to the rough equivalence between $\|f\|_{U^{2}}$ and $\|\hat{f}\|_{\infty}$ when $\|f\|_{\infty}\leq 1$. It can also be thought of as a soft inverse theorem; property (iii) then tells us that the functions that we obtain from this inverse theorem have products that are not too big.

Now let us briefly see why Theorem 4.3 generalizes easily from preduals of algebra norms to QAP-norms. The following result is not quite the result alluded to in the title of this subsection, but it is an abstract principle that can be used as part of the proof the Green-Tao theorem. As we mentioned in the introduction, the proof given here is much shorter and simpler than the proof given by Green and Tao. (This is not quite trivial to verify as they do not explicitly state the result, but the proof here can be used to simplify Section 6 of their paper slightly, and to replace Sections 7 and 8 completely.)

It is immediate from the definition of $\rho^{\prime}(C_{1},C_{2},\delta,J)$ that for every $C_{1}$, $C_{2}$ and $\delta$ there exists a polynomial $P$ such that $|P(x)-J(x)|\leq\delta$ for every $x\in[-C_{1},C_{1}]$, and such that $R_{P}^{\prime}(C_{2})\leq\rho^{\prime}(C_{1},C_{2},\delta,J)$.

From the formula for $\|\phi\|_{BAC}^{*}$ and the fact that this is at most $C_{2}$ it follows that for any $\epsilon>0$ we can write $\phi$ as a linear combination of basic anti-uniform functions, with the absolute values of the coefficients adding up to at most $C_{2}+\epsilon$. Therefore, for any $\epsilon>0$ we can write $\phi^{m}$ as a linear combination of products of $m$ basic anti-uniform functions, with the absolute values of the coefficients adding up to at most $C_{2}^{m}+\epsilon$. Each of these products has $\|.\|^{*}$-norm at most $C(m)$, by property (iii) of QAP-norms. Hence, by the triangle inequality, $\|\phi^{m}\|^{*}\leq C(m)C_{2}^{m}$. More generally, if $P$ is the polynomial $P(x)=a_{n}x^{n}+\dots+a_{1}x+a_{0}$, then by the triangle inequality we obtain that