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The purpose of this post is to isolate a combinatorial optimisation problem regarding subset sums; any improvement upon the current known bounds for this problem would lead to numerical improvements for the quantities pursued in the Polymath8 project. (UPDATE: Unfortunately no purely combinatorial improvement is possible, see comments.) We will also record the number-theoretic details of how this combinatorial problem is used in Zhang’s argument establishing bounded prime gaps.

First, some (rough) motivational background, omitting all the number-theoretic details and focusing on the combinatorics. (But readers who just want to see the combinatorial problem can skip the motivation and jump ahead to Lemma 5.) As part of the Polymath8 project we are trying to establish a certain estimate called {MPZ[\varpi,\delta]} for as wide a range of {\varpi,\delta > 0} as possible. Currently the best result we have is:

Theorem 1 (Zhang’s theorem, numerically optimised) {MPZ[\varpi,\delta]} holds whenever {207\varpi + 43\delta< \frac{1}{4}}.

Enlarging this region would lead to a better value of certain parameters {k_0}, {H} which in turn control the best bound on asymptotic gaps between consecutive primes. See this previous post for more discussion of this. At present, the best value {k_0=23,283} of {k_0} is applied by taking {(\varpi,\delta)} sufficiently close to {(1/899,71/154628)}, so improving Theorem 1 in the neighbourhood of this value is particularly desirable.

I’ll state exactly what {MPZ[\varpi,\delta]} is below the fold. For now, suffice to say that it involves a certain number-theoretic function, the von Mangoldt function {\Lambda}. To prove the theorem, the first step is to use a certain identity (the Heath-Brown identity) to decompose {\Lambda} into a lot of pieces, which take the form

\displaystyle  \alpha_{1} \ast \ldots \ast \alpha_{n} \ \ \ \ \ (1)

for some bounded {n} (in Zhang’s paper {n} never exceeds {20}) and various weights {\alpha_{1},\ldots,\alpha_n} supported at various scales {N_1,\ldots,N_n \geq 1} that multiply up to approximately {x}:

\displaystyle  N_1 \ldots N_n \sim x.

We can write {N_i = x^{t_i}}, thus ignoring negligible errors, {t_1,\ldots,t_n} are non-negative real numbers that add up to {1}:

\displaystyle  t_1 + \ldots + t_n = 1.

A key technical feature of the Heath-Brown identity is that the weight {\alpha_i} associated to sufficiently large values of {t_i} (e.g. {t_i \geq 1/10}) are “smooth” in a certain sense; this will be detailed below the fold.

The operation {\ast} is Dirichlet convolution, which is commutative and associative. We can thus regroup the convolution (1) in a number of ways. For instance, given any partition {\{1,\ldots,n\} = S \cup T} into disjoint sets {S,T}, we can rewrite (1) as

\displaystyle  \alpha_S \ast \alpha_T

where {\alpha_S} is the convolution of those {\alpha_i} with {i \in S}, and similarly for {\alpha_T}.

Zhang’s argument splits into two major pieces, in which certain classes of (1) are established. Cheating a little bit, the following three results are established:

Theorem 2 (Type 0 estimate, informal version) The term (1) gives an acceptable contribution to {MPZ[\varpi,\delta]} whenever

\displaystyle  t_i > \frac{1}{2} + 2 \varpi

for some {i}.

Theorem 3 (Type I/II estimate, informal version) The term (1) gives an acceptable contribution to {MPZ[\varpi,\delta]} whenever one can find a partition {\{1,\ldots,n\} = S \cup T} such that

\displaystyle  \frac{1}{2} - \sigma < \sum_{i \in S} t_i \leq \sum_{i \in T} t_i < \frac{1}{2} + \sigma

where {\sigma} is a quantity such that

\displaystyle  11 \varpi + 3\delta + 2 \sigma < \frac{1}{4}.

Theorem 4 (Type III estimate, informal version) The term (1) gives an acceptable contribution to {MPZ[\varpi,\delta]} whenever one can find {t_i,t_j,t_k} with distinct {i,j,k \in \{1,\ldots,n\}} with

\displaystyle  t_i \leq t_j \leq t_k \leq \frac{1}{2}

and

\displaystyle  4t_k + 4t_j + 5t_i > 4 + 16 \varpi + \delta.

The above assertions are oversimplifications; there are some additional minor smallness hypotheses on {\varpi,\delta} that are needed but at the current (small) values of {\varpi,\delta} under consideration they are not relevant and so will be omitted.

The deduction of Theorem 1 from Theorems 2, 3, 4 is then accomplished from the following, purely combinatorial, lemma:

Lemma 5 (Subset sum lemma) Let {\varpi,\delta > 0} be such that

\displaystyle  207\varpi + 43\delta < \frac{1}{4}. \ \ \ \ \ (2)

Let {t_1,\ldots,t_n} be non-negative reals such that

\displaystyle  t_1 + \ldots + t_n = 1.

Then at least one of the following statements hold:

  • (Type 0) There is {1 \leq i \leq n} such that {t_i > \frac{1}{2} + 2 \varpi}.
  • (Type I/II) There is a partition {\{1,\ldots,n\} = S \cup T} such that

    \displaystyle  \frac{1}{2} - \sigma < \sum_{i \in S} t_i \leq \sum_{i \in T} t_i < \frac{1}{2} + \sigma

    where {\sigma} is a quantity such that

    \displaystyle  11 \varpi + 3\delta + 2 \sigma < \frac{1}{4}.

  • (Type III) One can find distinct {t_i,t_j,t_k} with

    \displaystyle  t_i \leq t_j \leq t_k \leq \frac{1}{2}

    and

    \displaystyle  4t_k + 4t_j + 5t_i > 4 + 16 \varpi + \delta.

The purely combinatorial question is whether the hypothesis (2) can be relaxed here to a weaker condition. This would allow us to improve the ranges for Theorem 1 (and hence for the values of {k_0} and {H} alluded to earlier) without needing further improvement on Theorems 2, 3, 4 (although such improvement is also going to be a focus of Polymath8 investigations in the future).

Let us review how this lemma is currently proven. The key sublemma is the following:

Lemma 6 Let {1/10 < \sigma < 1/2}, and let {t_1,\ldots,t_n} be non-negative numbers summing to {1}. Then one of the following three statements hold:

  • (Type 0) There is a {t_i} with {t_i \geq 1/2 + \sigma}.
  • (Type I/II) There is a partition {\{1,\ldots,n\} = S \cup T} such that

    \displaystyle  \frac{1}{2} - \sigma < \sum_{i \in S} t_i \leq \sum_{i \in T} t_i < \frac{1}{2} + \sigma.

  • (Type III) There exist distinct {i,j,k} with {2\sigma \leq t_i \leq t_j \leq t_k \leq 1/2-\sigma} and {t_i+t_j,t_i+t_k,t_j+t_k \geq 1/2 + \sigma}.

Proof: Suppose Type I/II never occurs, then every partial sum {\sum_{i \in S} t_i} is either “small” in the sense that it is less than or equal to {1/2-\sigma}, or “large” in the sense that it is greater than or equal to {1/2+\sigma}, since otherwise we would be in the Type I/II case either with {S} as is and {T} the complement of {S}, or vice versa.

Call a summand {t_i} “powerless” if it cannot be used to turn a small partial sum into a large partial sum, thus there are no {S \subset \{1,\ldots,n\} \backslash \{i\}} such that {\sum_{j \in S} t_j} is small and {t_i + \sum_{j \in S} t_j} is large. We then split {\{1,\ldots,n\} = A \cup B} where {A} are the powerless elements and {B} are the powerful elements.

By induction we see that if {S \subset B} and {\sum_{i \in S} t_i} is small, then {\sum_{i \in S} t_i + \sum_{i \in A} t_i} is also small. Thus every sum of powerful summand is either less than {1/2-\sigma-\sum_{i \in A} t_i} or larger than {1/2+\sigma}. Since a powerful element must be able to convert a small sum to a large sum (in fact it must be able to convert a small sum of powerful summands to a large sum, by stripping out the powerless summands), we conclude that every powerful element has size greater than {2\sigma + \sum_{i \in A} t_i}. We may assume we are not in Type 0, then every powerful summand is at least {2\sigma + \sum_{i \in A} t_i} and at most {1/2 - \sigma - \sum_{i \in A} t_i}. In particular, there have to be at least three powerful summands, otherwise {\sum_{i=1}^n t_i} cannot be as large as {1}. As {\sigma > 1/10}, we have {4\sigma > 1/2-\sigma}, and we conclude that the sum of any two powerful summands is large (which, incidentally, shows that there are exactly three powerful summands). Taking {t_i,t_j,t_k} to be three powerful summands in increasing order we land in Type III. \Box

Now we see how Lemma 6 implies Lemma 5. Let {\varpi,\delta} be as in Lemma 5. We take {\sigma} almost as large as we can for the Type I/II case, thus we set

\displaystyle  \sigma := \frac{1}{8} - \frac{11}{2} \varpi - \frac{3}{2} \delta - \epsilon \ \ \ \ \ (3)

for some sufficiently small {\epsilon>0}. We observe from (2) that we certainly have

\displaystyle  \sigma > 2 \varpi

and

\displaystyle  \sigma > \frac{1}{10}

with plenty of room to spare. We then apply Lemma 6. The Type 0 case of that lemma then implies the Type 0 case of Lemma 5, while the Type I/II case of Lemma 6 also implies the Type I/II case of Lemma 5. Finally, suppose that we are in the Type III case of Lemma 6. Since

\displaystyle  4t_i + 4t_j + 5 t_k = \frac{5}{2} (t_i+t_k) + \frac{5}{2}(t_j+t_k) + \frac{3}{2} (t_i+t_j)

we thus have

\displaystyle  4t_i + 4t_j + 5 t_k \geq \frac{13}{2} (\frac{1}{2}+\sigma)

and so we will be done if

\displaystyle  \frac{13}{2} (\frac{1}{2}+\sigma) > 4 + 16 \varpi + \delta.

Inserting (3) and taking {\epsilon} small enough, it suffices to verify that

\displaystyle  \frac{13}{2} (\frac{1}{2}+\frac{1}{8} - \frac{11}{2} \varpi - \frac{3}{2}\delta) > 4 + 16 \varpi + \delta

but after some computation this is equivalent to (2).

It seems that there is some slack in this computation; some of the conclusions of the Type III case of Lemma 5, in particular, ended up being “wasted”, and it is possible that one did not fully exploit all the partial sums that could be used to create a Type I/II situation. So there may be a way to make improvements through purely combinatorial arguments. (UPDATE: As it turns out, this is sadly not the case: consderation of the case when {n=4}, {t_1 = 1/4 - 3\sigma/2}, and {t_2=t_3=t_4 = 1/4+\sigma/2} shows that one cannot obtain any further improvement without actually improving the Type I/II and Type III analysis.)

A technical remark: for the application to Theorem 1, it is possible to enforce a bound on the number {n} of summands in Lemma 5. More precisely, we may assume that {n} is an even number of size at most {n \leq 2K} for any natural number {K} we please, at the cost of adding the additioal constraint {t_i > \frac{1}{K}} to the Type III conclusion. Since {t_i} is already at least {2\sigma}, which is at least {\frac{1}{5}}, one can safely take {K=5}, so {n} can be taken to be an even number of size at most {10}, which in principle makes the problem of optimising Lemma 5 a fixed linear programming problem. (Zhang takes {K=10}, but this appears to be overkill. On the other hand, {K} does not appear to be a parameter that overly influences the final numerical bounds.)

Below the fold I give the number-theoretic details of the combinatorial aspects of Zhang’s argument that correspond to the combinatorial problem described above.

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One of the basic objects of study in combinatorics are finite strings {(a_n)_{n=0}^N} or infinite strings {(a_n)_{n=0}^\infty} of symbols {a_n} from some given alphabet {{\mathcal A}}, which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set {A} of natural numbers can be identified with the infinite string {(1_A(n))_{n=0}^\infty} of {0}s and {1}s formed by the indicator of {A}, e.g. the even numbers can be identified with the string {1010101\ldots} from the alphabet {\{0,1\}}, the multiples of three can be identified with the string {100100100\ldots}, and so forth. One can also consider doubly infinite strings {(a_n)_{n \in {\bf Z}}}, which among other things can be used to describe arbitrary subsets of integers.

On the other hand, the basic object of study in dynamics (and in related fields, such as ergodic theory) is that of a dynamical system {(X,T)}, that is to say a space {X} together with a shift map {T: X \rightarrow X} (which is often assumed to be invertible, although one can certainly study non-invertible dynamical systems as well). One often adds additional structure to this dynamical system, such as topological structure (giving rise topological dynamics), measure-theoretic structure (giving rise to ergodic theory), complex structure (giving rise to complex dynamics), and so forth. A dynamical system gives rise to an action of the natural numbers {{\bf N}} on the space {X} by using the iterates {T^n: X \rightarrow X} of {T} for {n=0,1,2,\ldots}; if {T} is invertible, we can extend this action to an action of the integers {{\bf Z}} on the same space. One can certainly also consider dynamical systems whose underlying group (or semi-group) is something other than {{\bf N}} or {{\bf Z}} (e.g. one can consider continuous dynamical systems in which the evolution group is {{\bf R}}), but we will restrict attention to the classical situation of {{\bf N}} or {{\bf Z}} actions here.

There is a fundamental correspondence principle connecting the study of strings (or subsets of natural numbers or integers) with the study of dynamical systems. In one direction, given a dynamical system {(X,T)}, an observable {c: X \rightarrow {\mathcal A}} taking values in some alphabet {{\mathcal A}}, and some initial datum {x_0 \in X}, we can first form the forward orbit {(T^n x_0)_{n=0}^\infty} of {x_0}, and then observe this orbit using {c} to obtain an infinite string {(c(T^n x_0))_{n=0}^\infty}. If the shift {T} in this system is invertible, one can extend this infinite string into a doubly infinite string {(c(T^n x_0))_{n \in {\bf Z}}}. Thus we see that every quadruplet {(X,T,c,x_0)} consisting of a dynamical system {(X,T)}, an observable {c}, and an initial datum {x_0} creates an infinite string.

Example 1 If {X} is the three-element set {X = {\bf Z}/3{\bf Z}} with the shift map {Tx := x+1}, {c: {\bf Z}/3{\bf Z} \rightarrow \{0,1\}} is the observable that takes the value {1} at the residue class {0 \hbox{ mod } 3} and zero at the other two classes, and one starts with the initial datum {x_0 = 0 \hbox{ mod } 3}, then the observed string {(c(T^n x_0))_{n=0}^\infty} becomes the indicator {100100100\ldots} of the multiples of three.

In the converse direction, every infinite string {(a_n)_{n=0}^\infty} in some alphabet {{\mathcal A}} arises (in a decidedly non-unique fashion) from a quadruple {(X,T,c,x_0)} in the above fashion. This can be easily seen by the following “universal” construction: take {X} to be the set {X:= {\mathcal A}^{\bf N}} of infinite strings {(b_i)_{n=0}^\infty} in the alphabet {{\mathcal A}}, let {T: X \rightarrow X} be the shift map

\displaystyle  T(b_i)_{n=0}^\infty := (b_{i+1})_{n=0}^\infty,

let {c: X \rightarrow {\mathcal A}} be the observable

\displaystyle  c((b_i)_{n=0}^\infty) := b_0,

and let {x_0 \in X} be the initial point

\displaystyle  x_0 := (a_i)_{n=0}^\infty.

Then one easily sees that the observed string {(c(T^n x_0))_{n=0}^\infty} is nothing more than the original string {(a_n)_{n=0}^\infty}. Note also that this construction can easily be adapted to doubly infinite strings by using {{\mathcal A}^{\bf Z}} instead of {{\mathcal A}^{\bf N}}, at which point the shift map {T} now becomes invertible. An important variant of this construction also attaches an invariant probability measure to {X} that is associated to the limiting density of various sets associated to the string {(a_i)_{n=0}^\infty}, and leads to the Furstenberg correspondence principle, discussed for instance in these previous blog posts. Such principles allow one to rigorously pass back and forth between the combinatorics of strings and the dynamics of systems; for instance, Furstenberg famously used his correspondence principle to demonstrate the equivalence of Szemerédi’s theorem on arithmetic progressions with what is now known as the Furstenberg multiple recurrence theorem in ergodic theory.

In the case when the alphabet {{\mathcal A}} is the binary alphabet {\{0,1\}}, and (for technical reasons related to the infamous non-injectivity {0.999\ldots = 1.00\ldots} of the decimal representation system) the string {(a_n)_{n=0}^\infty} does not end with an infinite string of {1}s, then one can reformulate the above universal construction by taking {X} to be the interval {[0,1)}, {T} to be the doubling map {Tx := 2x \hbox{ mod } 1}, {c: X \rightarrow \{0,1\}} to be the observable that takes the value {1} on {[1/2,1)} and {0} on {[0,1/2)} (that is, {c(x)} is the first binary digit of {x}), and {x_0} is the real number {x_0 := \sum_{n=0}^\infty a_n 2^{-n-1}} (that is, {x_0 = 0.a_0a_1\ldots} in binary).

The above universal construction is very easy to describe, and is well suited for “generic” strings {(a_n)_{n=0}^\infty} that have no further obvious structure to them, but it often leads to dynamical systems that are much larger and more complicated than is actually needed to produce the desired string {(a_n)_{n=0}^\infty}, and also often obscures some of the key dynamical features associated to that sequence. For instance, to generate the indicator {100100100\ldots} of the multiples of three that were mentioned previously, the above universal construction requires an uncountable space {X} and a dynamics which does not obviously reflect the key features of the sequence such as its periodicity. (Using the unit interval model, the dynamics arise from the orbit of {2/7} under the doubling map, which is a rather artificial way to describe the indicator function of the multiples of three.)

A related aesthetic objection to the universal construction is that of the four components {X,T,c,x_0} of the quadruplet {(X,T,c,x_0)} used to generate the sequence {(a_n)_{n=0}^\infty}, three of the components {X,T,c} are completely universal (in that they do not depend at all on the sequence {(a_n)_{n=0}^\infty}), leaving only the initial datum {x_0} to carry all the distinctive features of the original sequence. While there is nothing wrong with this mathematically, from a conceptual point of view it would make sense to make all four components of the quadruplet to be adapted to the sequence, in order to take advantage of the accumulated intuition about various special dynamical systems (and special observables), not just special initial data.

One step in this direction can be made by restricting {X} to the orbit {\{ T^n x_0: n \in {\bf N} \}} of the initial datum {x_0} (actually for technical reasons it is better to restrict to the topological closure {\overline{\{ T^n x_0: n \in {\bf N} \}}} of this orbit, in order to keep {X} compact). For instance, starting with the sequence {100100100\ldots}, the orbit now consists of just three points {100100100\ldots}, {010010010\ldots}, {001001001\ldots}, bringing the system more in line with the example in Example 1. Technically, this is the “optimal” representation of the sequence by a quadruplet {(X,T,c,x_0)}, because any other such representation {(X',T',c',x'_0)} is a factor of this representation (in the sense that there is a unique map {\pi: X \rightarrow X'} with {T' \circ \pi = \pi \circ T}, {c' \circ \pi = c}, and {x'_0 = \pi(x_0)}). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system {X} are interpreted as infinite strings rather than elements of a more geometrically or algebraically rich object (e.g. points in a circle, torus, or other homogeneous space).

For general sequences {(a_n)_{n=0}^\infty}, locating relevant geometric or algebraic structure in a dynamical system generating that sequence is an important but very difficult task (see e.g. this paper of Host and Kra, which is more or less devoted to precisely this task in the context of working out what component of a dynamical system controls the multiple recurrence behaviour of that system). However, for specific examples of sequences {(a_n)_{n=0}^\infty}, one can use an informal procedure of educated guesswork in order to produce a more natural-looking quadruple {(X,T,c,x_0)} that generates that sequence. This is not a particularly difficult or deep operation, but I found it very helpful in internalising the intuition behind the correspondence principle. Being non-rigorous, this procedure does not seem to be emphasised in most presentations of the correspondence principle, so I thought I would describe it here.

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The rectification principle in arithmetic combinatorics asserts, roughly speaking, that very small subsets (or, alternatively, small structured subsets) of an additive group or a field of large characteristic can be modeled (for the purposes of arithmetic combinatorics) by subsets of a group or field of zero characteristic, such as the integers {{\bf Z}} or the complex numbers {{\bf C}}. The additive form of this principle is known as the Freiman rectification principle; it has several formulations, going back of course to the original work of Freiman. Here is one formulation as given by Bilu, Lev, and Ruzsa:

Proposition 1 (Additive rectification) Let {A} be a subset of the additive group {{\bf Z}/p{\bf Z}} for some prime {p}, and let {s \geq 1} be an integer. Suppose that {|A| \leq \log_{2s} p}. Then there exists a map {\phi: A \rightarrow A'} into a subset {A'} of the integers which is a Freiman isomorphism of order {s} in the sense that for any {x_1,\ldots,x_s,y_1,\ldots,y_s \in A}, one has

\displaystyle  x_1+\ldots+x_s = y_1+\ldots+y_s

if and only if

\displaystyle  \phi(x_1)+\ldots+\phi(x_s) = \phi(y_1)+\ldots+\phi(y_s).

Furthermore {\phi} is a right-inverse of the obvious projection homomorphism from {{\bf Z}} to {{\bf Z}/p{\bf Z}}.

The original version of the rectification principle allowed the sets involved to be substantially larger in size (cardinality up to a small constant multiple of {p}), but with the additional hypothesis of bounded doubling involved; see the above-mentioned papers, as well as this later paper of Green and Ruzsa, for further discussion.

The proof of Proposition 1 is quite short (see Theorem 3.1 of Bilu-Lev-Ruzsa); the main idea is to use Minkowski’s theorem to find a non-trivial dilate {aA} of {A} that is contained in a small neighbourhood of the origin in {{\bf Z}/p{\bf Z}}, at which point the rectification map {\phi} can be constructed by hand.

Very recently, Codrut Grosu obtained an arithmetic analogue of the above theorem, in which the rectification map {\phi} preserves both additive and multiplicative structure:

Theorem 2 (Arithmetic rectification) Let {A} be a subset of the finite field {{\bf F}_p} for some prime {p \geq 3}, and let {s \geq 1} be an integer. Suppose that {|A| < \log_2 \log_{2s} \log_{2s^2} p - 1}. Then there exists a map {\phi: A \rightarrow A'} into a subset {A'} of the complex numbers which is a Freiman field isomorphism of order {s} in the sense that for any {x_1,\ldots,x_n \in A} and any polynomial {P(x_1,\ldots,x_n)} of degree at most {s} and integer coefficients of magnitude summing to at most {s}, one has

\displaystyle  P(x_1,\ldots,x_n)=0

if and only if

\displaystyle  P(\phi(x_1),\ldots,\phi(x_n))=0.

Note that it is necessary to use an algebraically closed field such as {{\bf C}} for this theorem, in contrast to the integers used in Proposition 1, as {{\bf F}_p} can contain objects such as square roots of {-1} which can only map to {\pm i} in the complex numbers (once {s} is at least {2}).

Using Theorem 2, one can transfer results in arithmetic combinatorics (e.g. sum-product or Szemerédi-Trotter type theorems) regarding finite subsets of {{\bf C}} to analogous results regarding sufficiently small subsets of {{\bf F}_p}; see the paper of Grosu for several examples of this. This should be compared with the paper of Vu, Wood, and Wood, which introduces a converse principle that embeds finite subsets of {{\bf C}} (or more generally, a characteristic zero integral domain) in a Freiman field-isomorphic fashion into finite subsets of {{\bf F}_p} for arbitrarily large primes {p}, allowing one to transfer arithmetic combinatorical facts from the latter setting to the former.

Grosu’s argument uses some quantitative elimination theory, and in particular a quantitative variant of a lemma of Chang that was discussed previously on this blog. In that previous blog post, it was observed that (an ineffective version of) Chang’s theorem could be obtained using only qualitative algebraic geometry (as opposed to quantitative algebraic geometry tools such as elimination theory results with explicit bounds) by means of nonstandard analysis (or, in what amounts to essentially the same thing in this context, the use of ultraproducts). One can then ask whether one can similarly establish an ineffective version of Grosu’s result by nonstandard means. The purpose of this post is to record that this can indeed be done without much difficulty, though the result obtained, being ineffective, is somewhat weaker than that in Theorem 2. More precisely, we obtain

Theorem 3 (Ineffective arithmetic rectification) Let {s, n \geq 1}. Then if {{\bf F}} is a field of characteristic at least {C_{s,n}} for some {C_{s,n}} depending on {s,n}, and {A} is a subset of {{\bf F}} of cardinality {n}, then there exists a map {\phi: A \rightarrow A'} into a subset {A'} of the complex numbers which is a Freiman field isomorphism of order {s}.

Our arguments will not provide any effective bound on the quantity {C_{s,n}} (though one could in principle eventually extract such a bound by deconstructing the proof of Proposition 4 below), making this result weaker than Theorem 2 (save for the minor generalisation that it can handle fields of prime power order as well as fields of prime order as long as the characteristic remains large).

Following the principle that ultraproducts can be used as a bridge to connect quantitative and qualitative results (as discussed in these previous blog posts), we will deduce Theorem 3 from the following (well-known) qualitative version:

Proposition 4 (Baby Lefschetz principle) Let {k} be a field of characteristic zero that is finitely generated over the rationals. Then there is an isomorphism {\phi: k \rightarrow \phi(k)} from {k} to a subfield {\phi(k)} of {{\bf C}}.

This principle (first laid out in an appendix of Lefschetz’s book), among other things, often allows one to use the methods of complex analysis (e.g. Riemann surface theory) to study many other fields of characteristic zero. There are many variants and extensions of this principle; see for instance this MathOverflow post for some discussion of these. I used this baby version of the Lefschetz principle recently in a paper on expanding polynomial maps.

Proof: We give two proofs of this fact, one using transcendence bases and the other using Hilbert’s nullstellensatz.

We begin with the former proof. As {k} is finitely generated over {{\bf Q}}, it has finite transcendence degree, thus one can find algebraically independent elements {x_1,\ldots,x_m} of {k} over {{\bf Q}} such that {k} is a finite extension of {{\bf Q}(x_1,\ldots,x_m)}, and in particular by the primitive element theorem {k} is generated by {{\bf Q}(x_1,\ldots,x_m)} and an element {\alpha} which is algebraic over {{\bf Q}(x_1,\ldots,x_m)}. (Here we use the fact that characteristic zero fields are separable.) If we then define {\phi} by first mapping {x_1,\ldots,x_m} to generic (and thus algebraically independent) complex numbers {z_1,\ldots,z_m}, and then setting {\phi(\alpha)} to be a complex root of of the minimal polynomial for {\alpha} over {{\bf Q}(x_1,\ldots,x_m)} after replacing each {x_i} with the complex number {z_i}, we obtain a field isomorphism {\phi: k \rightarrow \phi(k)} with the required properties.

Now we give the latter proof. Let {x_1,\ldots,x_m} be elements of {k} that generate that field over {{\bf Q}}, but which are not necessarily algebraically independent. Our task is then equivalent to that of finding complex numbers {z_1,\ldots,z_m} with the property that, for any polynomial {P(x_1,\ldots,x_m)} with rational coefficients, one has

\displaystyle  P(x_1,\ldots,x_m) = 0

if and only if

\displaystyle  P(z_1,\ldots,z_m) = 0.

Let {{\mathcal P}} be the collection of all polynomials {P} with rational coefficients with {P(x_1,\ldots,x_m)=0}, and {{\mathcal Q}} be the collection of all polynomials {P} with rational coefficients with {P(x_1,\ldots,x_m) \neq 0}. The set

\displaystyle  S := \{ (z_1,\ldots,z_m) \in {\bf C}^m: P(z_1,\ldots,z_m)=0 \hbox{ for all } P \in {\mathcal P} \}

is the intersection of countably many algebraic sets and is thus also an algebraic set (by the Hilbert basis theorem or the Noetherian property of algebraic sets). If the desired claim failed, then {S} could be covered by the algebraic sets {\{ (z_1,\ldots,z_m) \in {\bf C}^m: Q(z_1,\ldots,z_m) = 0 \}} for {Q \in {\mathcal Q}}. By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over {{\bf C}} cannot be covered by countably many varieties of smaller dimension, we conclude that {S} must in fact be covered by a finite number of such sets, thus

\displaystyle  S \subset \bigcup_{i=1}^n \{ (z_1,\ldots,z_m) \in {\bf C}^m: Q_i(z_1,\ldots,z_m) = 0 \}

for some {Q_1,\ldots,Q_n \in {\bf C}^m}. By the nullstellensatz, we thus have an identity of the form

\displaystyle  (Q_1 \ldots Q_n)^l = P_1 R_1 + \ldots + P_r R_r

for some natural numbers {l,r \geq 1}, polynomials {P_1,\ldots,P_r \in {\mathcal P}}, and polynomials {R_1,\ldots,R_r} with coefficients in {\overline{{\bf Q}}}. In particular, this identity also holds in the algebraic closure {\overline{k}} of {k}. Evaluating this identity at {(x_1,\ldots,x_m)} we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows. \Box

From Proposition 4 one can now deduce Theorem 3 by a routine ultraproduct argument (the same one used in these previous blog posts). Suppose for contradiction that Theorem 3 fails. Then there exists natural numbers {s,n \geq 1}, a sequence of finite fields {{\bf F}_i} of characteristic at least {i}, and subsets {A_i=\{a_{i,1},\ldots,a_{i,n}\}} of {{\bf F}_i} of cardinality {n} such that for each {i}, there does not exist a Freiman field isomorphism of order {s} from {A_i} to the complex numbers. Now we select a non-principal ultrafilter {\alpha \in \beta {\bf N} \backslash {\bf N}}, and construct the ultraproduct {{\bf F} := \prod_{i \rightarrow \alpha} {\bf F}_i} of the finite fields {{\bf F}_i}. This is again a field (and is a basic example of what is known as a pseudo-finite field); because the characteristic of {{\bf F}_i} goes to infinity as {i \rightarrow \infty}, it is easy to see (using Los’s theorem) that {{\bf F}} has characteristic zero and can thus be viewed as an extension of the rationals {{\bf Q}}.

Now let {a_j := \lim_{i \rightarrow \alpha} a_{i,j}} be the ultralimit of the {a_{i,j}}, so that {A := \{a_1,\ldots,a_n\}} is the ultraproduct of the {A_i}, then {A} is a subset of {{\bf F}} of cardinality {n}. In particular, if {k} is the field generated by {{\bf Q}} and {A}, then {k} is a finitely generated extension of the rationals and thus, by Proposition 4 there is an isomorphism {\phi: k \rightarrow \phi(k)} from {k} to a subfield {\phi(k)} of the complex numbers. In particular, {\phi(a_1),\ldots,\phi(a_n)} are complex numbers, and for any polynomial {P(x_1,\ldots,x_n)} with integer coefficients, one has

\displaystyle  P(a_1,\ldots,a_n) = 0

if and only if

\displaystyle  P(\phi(a_1),\ldots,\phi(a_n)) = 0.

By Los’s theorem, we then conclude that for all {i} sufficiently close to {\alpha}, one has for all polynomials {P(x_1,\ldots,x_n)} of degree at most {s} and whose coefficients are integers whose magnitude sums up to {s}, one has

\displaystyle  P(a_{i,1},\ldots,a_{i,n}) = 0

if and only if

\displaystyle  P(\phi(a_1),\ldots,\phi(a_n)) = 0.

But this gives a Freiman field isomorphism of order {s} between {A_i} and {\phi(A)}, contradicting the construction of {A_i}, and Theorem 3 follows.

The following result is due independently to Furstenberg and to Sarkozy:

Theorem 1 (Furstenberg-Sarkozy theorem) Let {\delta > 0}, and suppose that {N} is sufficiently large depending on {\delta}. Then every subset {A} of {[N] := \{1,\ldots,N\}} of density {|A|/N} at least {\delta} contains a pair {n, n+r^2} for some natural numbers {n, r} with {r \neq 0}.

This theorem is of course similar in spirit to results such as Roth’s theorem or Szemerédi’s theorem, in which the pattern {n,n+r^2} is replaced by {n,n+r,n+2r} or {n,n+r,\ldots,n+(k-1)r} for some fixed {k} respectively. There are by now many proofs of this theorem (see this recent paper of Lyall for a survey), but most proofs involve some form of Fourier analysis (or spectral theory). This may be compared with the standard proof of Roth’s theorem, which combines some Fourier analysis with what is now known as the density increment argument.

A few years ago, Ben Green, Tamar Ziegler, and myself observed that it is possible to prove the Furstenberg-Sarkozy theorem by just using the Cauchy-Schwarz inequality (or van der Corput lemma) and the density increment argument, removing all invocations of Fourier analysis, and instead relying on Cauchy-Schwarz to linearise the quadratic shift {r^2}. As such, this theorem can be considered as even more elementary than Roth’s theorem (and its proof can be viewed as a toy model for the proof of Roth’s theorem). We ended up not doing too much with this observation, so decided to share it here.

The first step is to use the density increment argument that goes back to Roth. For any {\delta > 0}, let {P(\delta)} denote the assertion that for {N} sufficiently large, all sets {A \subset [N]} of density at least {\delta} contain a pair {n,n+r^2} with {r} non-zero. Note that {P(\delta)} is vacuously true for {\delta > 1}. We will show that for any {0 < \delta_0 \leq 1}, one has the implication

\displaystyle  P(\delta_0 + c \delta_0^3) \implies P(\delta_0) \ \ \ \ \ (1)

for some absolute constant {c>0}. This implies that {P(\delta)} is true for any {\delta>0} (as can be seen by considering the infimum of all {\delta>0} for which {P(\delta)} holds), which gives Theorem 1.

It remains to establish the implication (1). Suppose for sake of contradiction that we can find {0 < \delta_0 \leq 1} for which {P(\delta_0+c\delta^3_0)} holds (for some sufficiently small absolute constant {c>0}), but {P(\delta_0)} fails. Thus, we can find arbitrarily large {N}, and subsets {A} of {[N]} of density at least {\delta_0}, such that {A} contains no patterns of the form {n,n+r^2} with {r} non-zero. In particular, we have

\displaystyle  \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} 1_A(n) 1_A(n+(r+h)^2) = 0.

(The exact ranges of {r} and {h} are not too important here, and could be replaced by various other small powers of {N} if desired.)

Let {\delta := |A|/N} be the density of {A}, so that {\delta_0 \leq \delta \leq 1}. Observe that

\displaystyle  \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} 1_A(n) \delta 1_{[N]}(n+(r+h)^2) = \delta^2 + O(N^{-1/3})

\displaystyle  \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} \delta 1_{[N]}(n) \delta 1_{[N]}(n+(r+h)^2) = \delta^2 + O(N^{-1/3})

and

\displaystyle  \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} \delta 1_{[N]}(n) 1_A(n+(r+h)^2) = \delta^2 + O( N^{-1/3} ).

If we thus set {f := 1_A - \delta 1_{[N]}}, then

\displaystyle  \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} f(n) f(n+(r+h)^2) = -\delta^2 + O( N^{-1/3} ).

In particular, for {N} large enough,

\displaystyle  \mathop{\bf E}_{n \in [N]} |f(n)| \mathop{\bf E}_{r \in [N^{1/3}]} |\mathop{\bf E}_{h \in [N^{1/100}]} f(n+(r+h)^2)| \gg \delta^2.

On the other hand, one easily sees that

\displaystyle  \mathop{\bf E}_{n \in [N]} |f(n)|^2 = O(\delta)

and hence by the Cauchy-Schwarz inequality

\displaystyle  \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} |\mathop{\bf E}_{h \in [N^{1/100}]} f(n+(r+h)^2)|^2 \gg \delta^3

which we can rearrange as

\displaystyle  |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h,h' \in [N^{1/100}]} \mathop{\bf E}_{n \in [N]} f(n+(r+h)^2) f(n+(r+h')^2)| \gg \delta^3.

Shifting {n} by {(r+h)^2} we obtain (again for {N} large enough)

\displaystyle  |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h,h' \in [N^{1/100}]} \mathop{\bf E}_{n \in [N]} f(n) f(n+(h'-h)(2r+h'+h))| \gg \delta^3.

In particular, by the pigeonhole principle (and deleting the diagonal case {h=h'}, which we can do for {N} large enough) we can find distinct {h,h' \in [N^{1/100}]} such that

\displaystyle  |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{n \in [N]} f(n) f(n+(h'-h)(2r+h'+h))| \gg \delta^3,

so in particular

\displaystyle  \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+(h'-h)(2r+h'+h))| \gg \delta^3.

If we set {d := 2(h'-h)} and shift {n} by {(h'-h) (h'+h)}, we can simplify this (again for {N} large enough) as

\displaystyle  \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr)| \gg \delta^3. \ \ \ \ \ (2)

On the other hand, since

\displaystyle  \mathop{\bf E}_{n \in [N]} f(n) = 0

we have

\displaystyle  \mathop{\bf E}_{n \in [N]} f(n+dr) = O( N^{-2/3+1/100})

for any {r \in [N^{1/3}]}, and thus

\displaystyle  \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) = O( N^{-2/3+1/100}).

Averaging this with (2) we conclude that

\displaystyle  \mathop{\bf E}_{n \in [N]} \max( \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr), 0 ) \gg \delta^3.

In particular, by the pigeonhole principle we can find {n \in [N]} such that

\displaystyle  \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) \gg \delta^3,

or equivalently {A} has density at least {\delta+c'\delta^3} on the arithmetic progression {\{ n+dr: r \in [N^{1/3}]\}}, which has length {\lfloor N^{1/3}\rfloor } and spacing {d}, for some absolute constant {c'>0}. By partitioning this progression into subprogressions of spacing {d^2} and length {\lfloor N^{1/4}\rfloor} (plus an error set of size {O(N^{1/4})}, we see from the pigeonhole principle that we can find a progression {\{ n' + d^2 r': r' \in [N^{1/4}]\}} of length {\lfloor N^{1/4}\rfloor} and spacing {d^2} on which {A} has density at least {\delta + c\delta^3} (and hence at least {\delta_0+c\delta_0^3}) for some absolute constant {c>0}. If we then apply the induction hypothesis to the set

\displaystyle  A' := \{ r' \in [N^{1/4}]: n' + d^2 r' \in A \}

we conclude (for {N} large enough) that {A'} contains a pair {m, m+s^2} for some natural numbers {m,s} with {s} non-zero. This implies that {(n'+d^2 m), (n'+d^2 m) + (|d|s)^2} lie in {A}, a contradiction, establishing the implication (1).

A more careful analysis of the above argument reveals a more quantitative version of Theorem 1: for {N \geq 100} (say), any subset of {[N]} of density at least {C/(\log\log N)^{1/2}} for some sufficiently large absolute constant {C} contains a pair {n,n+r^2} with {r} non-zero. This is not the best bound known; a (difficult) result of Pintz, Steiger, and Szemeredi allows the density to be as low as {C / (\log N)^{\frac{1}{4} \log\log\log\log N}}. On the other hand, this already improves on the (simpler) Fourier-analytic argument of Green that works for densities at least {C/(\log\log N)^{1/11}} (although the original argument of Sarkozy, which is a little more intricate, works up to {C (\log\log N)^{2/3}/(\log N)^{1/3}}). In the other direction, a construction of Rusza gives a set of density {\frac{1}{65} N^{-0.267}} without any pairs {n,n+r^2}.

Remark 1 A similar argument also applies with {n,n+r^2} replaced by {n,n+r^k} for fixed {k}, because this sort of pattern is preserved by affine dilations {r' \mapsto n'+d^k r'} into arithmetic progressions whose spacing {d^k} is a {k^{th}} power. By re-introducing Fourier analysis, one can also perform an argument of this type for {n,n+d,n+2d} where {d} is the sum of two squares; see the above-mentioned paper of Green for details. However there seems to be some technical difficulty in extending it to patterns of the form {n,n+P(r)} for polynomials {P} that consist of more than a single monomial (and with the normalisation {P(0)=0}, to avoid local obstructions), because one no longer has this preservation property.

Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our survey “Small doubling in groups“, for the proceedings of the upcoming Erdos Centennial.  This is a short survey of the known results on classifying finite subsets A of an (abelian) additive group G = (G,+) or a (not necessarily abelian) multiplicative group G = (G,\cdot) that have small doubling in the sense that the sum set A+A or product set A \cdot A is small.  Such sets behave approximately like finite subgroups of G (and there is a closely related notion of an approximate group in which the analogy is even tighter) , and so this subject can be viewed as a sort of approximate version of finite group theory.  (Unfortunately, thus far the theory does not have much new to say about the classification of actual finite groups; progress has been largely made instead on classifying the (highly restricted) number of ways in which approximate groups can differ from a genuine group.)

In the classical case when G is the integers {\mathbb Z}, these sets were classified (in a qualitative sense, at least) by a celebrated theorem of Freiman, which roughly speaking says that such sets A are necessarily “commensurate” in some sense with a (generalised) arithmetic progression P of bounded rank.   There are a number of essentially equivalent ways to define what “commensurate” means here; for instance, in the original formulation of the theorem, one asks that A be a dense subset of P, but in modern formulations it is often more convenient to require instead that A be of comparable size to P and be covered by a bounded number of translates of P, or that A and P have an intersection that is of comparable size to both A and P (cf. the notion of commensurability in group theory).

Freiman’s original theorem was extended to more general abelian groups in a sequence of papers culminating in the paper of Green and Ruzsa that handled arbitrary abelian groups.   As such groups now contain non-trivial finite subgroups, the conclusion of the theorem must be  modified by allowing for “coset progressions” P+H, which can be viewed as “extensions”  of generalized arithmetic progressions P by genuine finite groups H.

The proof methods in these abelian results were Fourier-analytic in nature (except in the cases of sets of very small doubling, in which more combinatorial approaches can be applied, and there were also some geometric or combinatorial methods that gave some weaker structural results).  As such, it was a challenge to extend these results to nonabelian groups, although for various important special types of ambient group G (such as an linear group over a finite or infinite field) it turns out that one can use tools exploiting the special structure of those groups (e.g. for linear groups one would use tools from Lie theory and algebraic geometry) to obtain quite satisfactory results; see e.g. this survey of  Pyber and Szabo for the linear case.   When the ambient group G is completely arbitrary, it turns out the problem is closely related to the classical Hilbert’s fifth problem of determining the minimal requirements of a topological group in order for such groups to have Lie structure; this connection was first observed and exploited by Hrushovski, and then used by Breuillard, Green, and myself to obtain the analogue of Freiman’s theorem for an arbitrary nonabelian group.

This survey is too short to discuss in much detail the proof techniques used in these results (although the abelian case is discussed in this book of mine with Vu, and the nonabelian case discussed in this more recent book of mine), but instead focuses on the statements of the various known results, as well as some remaining open questions in the subject (in particular, there is substantial work left to be done in making the estimates more quantitative, particularly in the nonabelian setting).

I’ve just uploaded to the arXiv my paper “Mixing for progressions in non-abelian groups“, submitted to Forum of Mathematics, Sigma (which, along with sister publication Forum of Mathematics, Pi, has just opened up its online submission system). This paper is loosely related in subject topic to my two previous papers on polynomial expansion and on recurrence in quasirandom groups (with Vitaly Bergelson), although the methods here are rather different from those in those two papers. The starting motivation for this paper was a question posed in this foundational paper of Tim Gowers on quasirandom groups. In that paper, Gowers showed (among other things) that if {G} was a quasirandom group, patterns such as {(x,xg,xh, xgh)} were mixing in the sense that, for any four sets {A,B,C,D \subset G}, the number of such quadruples {(x,xg,xh, xgh)} in {A \times B \times C \times D} was equal to {(\mu(A) \mu(B) \mu(C) \mu(D) + o(1)) |G|^3}, where {\mu(A) := |A| / |G|}, and {o(1)} denotes a quantity that goes to zero as the quasirandomness of the group goes to infinity. In my recent paper with Vitaly, we also considered mixing properties of some other patterns, namely {(x,xg,gx)} and {(g,x,xg,gx)}. This paper is concerned instead with the pattern {(x,xg,xg^2)}, that is to say a geometric progression of length three. As observed by Gowers, by applying (a suitably quantitative version of) Roth’s theorem in (cosets of) a cyclic group, one can obtain a recurrence theorem for this pattern without much effort: if {G} is an arbitrary finite group, and {A} is a subset of {G} with {\mu(A) \geq \delta}, then there are at least {c(\delta) |G|^2} pairs {(x,g) \in G} such that {x, xg, xg^2 \in A}, where {c(\delta)>0} is a quantity depending only on {\delta}. However, this argument does not settle the question of whether there is a stronger mixing property, in that the number of pairs {(x,g) \in G^2} such that {(x,xg,xg^2) \in A \times B \times C} should be {(\mu(A)\mu(B)\mu(C)+o(1)) |G|^2} for any {A,B,C \subset G}. Informally, this would assert that for {x, g} chosen uniformly at random from {G}, the triplet {(x, xg, xg^2)} should resemble a uniformly selected element of {G^3} in some weak sense.

For non-quasirandom groups, such mixing properties can certainly fail. For instance, if {G} is the cyclic group {G = ({\bf Z}/N{\bf Z},+)} (which is abelian and thus highly non-quasirandom) with the additive group operation, and {A = \{1,\ldots,\lfloor \delta N\rfloor\}} for some small but fixed {\delta > 0}, then {\mu(A) = \delta + o(1)} in the limit {N \rightarrow \infty}, but the number of pairs {(x,g) \in G^2} with {x, x+g, x+2g \in A} is {(\delta^2/2 + o(1)) |G|^2} rather than {(\delta^3+o(1)) |G|^2}. The problem here is that the identity {(x+2g) = 2(x+g) - x} ensures that if {x} and {x+g} both lie in {A}, then {x+2g} has a highly elevated likelihood of also falling in {A}. One can view {A} as the preimage of a small ball under the one-dimensional representation {\rho: G \rightarrow U(1)} defined by {\rho(n) := e^{2\pi i n/N}}; similar obstructions to mixing can also be constructed from other low-dimensional representations.

However, by definition, quasirandom groups do not have low-dimensional representations, and Gowers asked whether mixing for {(x,xg,xg^2)} could hold for quasirandom groups. I do not know if this is the case for arbitrary quasirandom groups, but I was able to settle the question for a specific class of quasirandom groups, namely the special linear groups {G := SL_d(F)} over a finite field {F} in the regime where the dimension {d} is bounded (but is at least two) and {F} is large. Indeed, for such groups I can obtain a count of {(\mu(A) \mu(B) \mu(C) + O( |F|^{-\min(d-1,2)/8} )) |G|^2} for the number of pairs {(x,g) \in G^2} with {(x, xg, xg^2) \in A \times B \times C}. In fact, I have the somewhat stronger statement that there are {(\mu(A) \mu(B) \mu(C) \mu(D) + O( |F|^{-\min(d-1,2)/8} )) |G|^2} pairs {(x,g) \in G^2} with {(x,xg,xg^2,g) \in A \times B \times C \times D} for any {A,B,C,D \subset G}.

I was also able to obtain a partial result for the length four progression {(x,xg,xg^2, xg^3)} in the simpler two-dimensional case {G = SL_2(F)}, but I had to make the unusual restriction that the group element {g \in G} was hyperbolic in the sense that it was diagonalisable over the finite field {F} (as opposed to diagonalisable over the algebraic closure {\overline{F}} of that field); this amounts to the discriminant of the matrix being a quadratic residue, and this holds for approximately half of the elements of {G}. The result is then that for any {A,B,C,D \subset G}, one has {(\frac{1}{2} \mu(A) \mu(B) \mu(C) \mu(D) + o(1)) |G|^2} pairs {(x,g) \in G} with {g} hyperbolic and {(x,xg,xg^2,xg^3) \subset A \times B \times C \times D}. (Again, I actually show a slightly stronger statement in which {g} is restricted to an arbitrary subset {E} of hyperbolic elements.)

For the length three argument, the main tools used are the Cauchy-Schwarz inequality, the quasirandomness of {G}, and some algebraic geometry to ensure that a certain family of probability measures on {G} that are defined algebraically are approximately uniformly distributed. The length four argument is significantly more difficult and relies on a rather ad hoc argument involving, among other things, expander properties related to the work of Bourgain and Gamburd, and also a “twisted” version of an argument of Gowers that is used (among other things) to establish an inverse theorem for the {U^3} norm.

I give some details of these arguments below the fold.

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Perhaps the most important structural result about general large dense graphs is the Szemerédi regularity lemma. Here is a standard formulation of that lemma:

Lemma 1 (Szemerédi regularity lemma) Let {G = (V,E)} be a graph on {n} vertices, and let {\epsilon > 0}. Then there exists a partition {V = V_1 \cup \ldots \cup V_M} for some {M \leq M(\epsilon)} with the property that for all but at most {\epsilon M^2} of the pairs {1 \leq i \leq j \leq M}, the pair {V_i, V_j} is {\epsilon}-regular in the sense that

\displaystyle  | d( A, B ) - d( V_i, V_j ) | \leq \epsilon

whenever {A \subset V_i, B \subset V_j} are such that {|A| \geq \epsilon |V_i|} and {|B| \geq \epsilon |V_j|}, and {d(A,B) := |\{ (a,b) \in A \times B: \{a,b\} \in E \}|/|A| |B|} is the edge density between {A} and {B}. Furthermore, the partition is equitable in the sense that {||V_i| - |V_j|| \leq 1} for all {1 \leq i \leq j \leq M}.

There are many proofs of this lemma, which is actually not that difficult to establish; see for instance these previous blog posts for some examples. In this post I would like to record one further proof, based on the spectral decomposition of the adjacency matrix of {G}, which is essentially due to Frieze and Kannan. (Strictly speaking, Frieze and Kannan used a variant of this argument to establish a weaker form of the regularity lemma, but it is not difficult to modify the Frieze-Kannan argument to obtain the usual form of the regularity lemma instead. Some closely related spectral regularity lemmas were also developed by Szegedy.) I found recently (while speaking at the Abel conference in honour of this year’s laureate, Endre Szemerédi) that this particular argument is not as widely known among graph theory experts as I had thought, so I thought I would record it here.

For reasons of exposition, it is convenient to first establish a slightly weaker form of the lemma, in which one drops the hypothesis of equitability (but then has to weight the cells {V_i} by their magnitude when counting bad pairs):

Lemma 2 (Szemerédi regularity lemma, weakened variant) . Let {G = (V,E)} be a graph on {n} vertices, and let {\epsilon > 0}. Then there exists a partition {V = V_1 \cup \ldots \cup V_M} for some {M \leq M(\epsilon)} with the property that for all pairs {(i,j) \in \{1,\ldots,M\}^2} outside of an exceptional set {\Sigma}, one has

\displaystyle  | E(A,B) - d_{ij} |A| |B| | \ll \epsilon |V_i| |V_j| \ \ \ \ \ (1)

whenever {A \subset V_i, B \subset V_j}, for some real number {d_{ij}}, where {E(A,B) := |\{ (a,b) \in A \times B: \{a,b\} \in E \}|} is the number of edges between {A} and {B}. Furthermore, we have

\displaystyle  \sum_{(i,j) \in \Sigma} |V_i| |V_j| \ll \epsilon |V|^2. \ \ \ \ \ (2)

Let us now prove Lemma 2. We enumerate {V} (after relabeling) as {V = \{1,\ldots,n\}}. The adjacency matrix {T} of the graph {G} is then a self-adjoint {n \times n} matrix, and thus admits an eigenvalue decomposition

\displaystyle  T = \sum_{i=1}^n \lambda_i u_i^* u_i

for some orthonormal basis {u_1,\ldots,u_n} of {{\bf C}^n} and some eigenvalues {\lambda_1,\ldots,\lambda_n \in {\bf R}}, which we arrange in decreasing order of magnitude:

\displaystyle  |\lambda_1| \geq \ldots \geq |\lambda_n|.

We can compute the trace of {T^2} as

\displaystyle  \hbox{tr}(T^2) = \sum_{i=1}^n |\lambda_i|^2.

But we also have {\hbox{tr}(T^2) = 2|E| \leq n^2}, so

\displaystyle  \sum_{i=1}^n |\lambda_i|^2 \leq n^2. \ \ \ \ \ (3)

Among other things, this implies that

\displaystyle  |\lambda_i| \leq \frac{n}{\sqrt{i}} \ \ \ \ \ (4)

for all {i \geq 1}.

Let {F: {\bf N} \rightarrow {\bf N}} be a function (depending on {\epsilon}) to be chosen later, with {F(i) \geq i} for all {i}. Applying (3) and the pigeonhole principle (or the finite convergence principle, see this blog post), we can find {J \leq C(F,\epsilon)} such that

\displaystyle  \sum_{J \leq i < F(J)} |\lambda_i|^2 \leq \epsilon^3 n^2.

(Indeed, the bound on {J} is basically {F} iterated {1/\epsilon^3} times.) We can now split

\displaystyle  T = T_1 + T_2 + T_3, \ \ \ \ \ (5)

where {T_1} is the “structured” component

\displaystyle  T_1 := \sum_{i < J} \lambda_i u_i^* u_i, \ \ \ \ \ (6)

{T_2} is the “small” component

\displaystyle  T_2 := \sum_{J \leq i < F(J)} \lambda_i u_i^* u_i, \ \ \ \ \ (7)

and {T_3} is the “pseudorandom” component

\displaystyle  T_3 := \sum_{i > F(J)} \lambda_i u_i^* u_i. \ \ \ \ \ (8)

We now design a vertex partition to make {T_1} approximately constant on most cells. For each {i < J}, we partition {V} into {O_{J,\epsilon}(1)} cells on which {u_i} (viewed as a function from {V} to {{\bf C}}) only fluctuates by {O(\epsilon n^{-1/2} /J)}, plus an exceptional cell of size {O( \frac{\epsilon}{J} |V|)} coming from the values where {|u_i|} is excessively large (larger than {\sqrt{\frac{J}{\epsilon}} n^{-1/2}}). Combining all these partitions together, we can write {V = V_1 \cup \ldots \cup V_{M-1} \cup V_M} for some {M = O_{J,\epsilon}(1)}, where {V_M} has cardinality at most {\epsilon |V|}, and for all {1 \leq i \leq M-1}, the eigenfunctions {u_1,\ldots,u_{J-1}} all fluctuate by at most {O(\epsilon/J)}. In particular, if {1 \leq i,j \leq M-1}, then (by (4) and (6)) the entries of {T_1} fluctuate by at most {O(\epsilon)} on each block {V_i \times V_j}. If we let {d_{ij}} be the mean value of these entries on {V_i \times V_j}, we thus have

\displaystyle  1_B^* T_1 1_A = d_{ij} |A| |B| + O( \epsilon |V_i| |V_j| ) \ \ \ \ \ (9)

for any {1 \leq i,j \leq M-1} and {A \subset V_i, B \subset V_j}, where we view the indicator functions {1_A, 1_B} as column vectors of dimension {n}.

Next, we observe from (3) and (7) that {\hbox{tr} T_2^2 \leq \epsilon^3 n^2}. If we let {x_{ab}} be the coefficients of {T_2}, we thus have

\displaystyle  \sum_{a,b \in V} |x_{ab}|^2 \leq \epsilon^3 n^2

and hence by Markov’s inequality we have

\displaystyle  \sum_{a \in V_i} \sum_{b \in V_j} |x_{ab}|^2 \leq \epsilon^2 |V_i| |V_j| \ \ \ \ \ (10)

for all pairs {(i,j) \in \{1,\ldots,M-1\}^2} outside of an exceptional set {\Sigma_1} with

\displaystyle  \sum_{(i,j) \in \Sigma_1} |V_i| |V_j| \leq \epsilon |V|^2.

If {(i,j) \in \{1,\ldots,M-1\}^2} avoids {\Sigma_1}, we thus have

\displaystyle  1_B^* T_2 1_A = O( \epsilon |V_i| |V_j| ) \ \ \ \ \ (11)

for any {A \subset V_i, B \subset V_j}, by (10) and the Cauchy-Schwarz inequality.

Finally, to control {T_3} we see from (4) and (8) that {T_3} has an operator norm of at most {n/\sqrt{F(J)}}. In particular, we have from the Cauchy-Schwarz inequality that

\displaystyle  1_B^* T_3 1_A = O( n^2 / \sqrt{F(J)} ) \ \ \ \ \ (12)

for any {A, B \subset V}.

Let {\Sigma} be the set of all pairs {(i,j) \in \{1,\ldots,M\}^2} where either {(i,j) \in \Sigma_1}, {i = M}, {j=M}, or

\displaystyle  \min(|V_i|, |V_j|) \leq \frac{\epsilon}{M} n.

One easily verifies that (2) holds. If {(i,j) \in \{1,\ldots,M\}^2} is not in {\Sigma}, then by summing (9), (11), (12) and using (5), we see that

\displaystyle  1_B^* T 1_A = d_{ij} |A| |B| + O( \epsilon |V_i| |V_j| ) + O( n^2 / \sqrt{F(J)} ) \ \ \ \ \ (13)

for all {A \subset V_i, B \subset V_j}. The left-hand side is just {E(A,B)}. As {(i,j) \not \in \Sigma}, we have

\displaystyle  |V_i|, |V_j| > \frac{\epsilon}{M} n

and so (since {M = O_{J,\epsilon}(1)})

\displaystyle  n^2 / \sqrt{F(J)} \ll_{J,\epsilon} |V_i| |V_j| / \sqrt{F(J)}.

If we let {F} be a sufficiently rapidly growing function of {J} that depends on {\epsilon}, the second error term in (13) can be absorbed in the first, and (1) follows. This concludes the proof of Lemma 2.

To prove Lemma 1, one argues similarly (after modifying {\epsilon} as necessary), except that the initial partition {V_1,\ldots,V_M} of {V} constructed above needs to be subdivided further into equitable components (of size {\epsilon |V|/M+O(1)}), plus some remainder sets which can be aggregated into an exceptional component of size {O( \epsilon |V| )} (and which can then be redistributed amongst the other components to arrive at a truly equitable partition). We omit the details.

Remark 1 It is easy to verify that {F} needs to be growing exponentially in {J} in order for the above argument to work, which leads to tower-exponential bounds in the number of cells {M} in the partition. It was shown by Gowers that a tower-exponential bound is actually necessary here. By varying {F}, one basically obtains the strong regularity lemma first established by Alon, Fischer, Krivelevich, and Szegedy; in the opposite direction, setting {F(J) := J} essentially gives the weak regularity lemma of Frieze and Kannan.

Remark 2 If we specialise to a Cayley graph, in which {V = (V,+)} is a finite abelian group and {E = \{ (a,b): a-b \in A \}} for some (symmetric) subset {A} of {V}, then the eigenvectors are characters, and one essentially recovers the arithmetic regularity lemma of Green, in which the vertex partition classes {V_i} are given by Bohr sets (and one can then place additional regularity properties on these Bohr sets with some additional arguments). The components {T_1, T_2, T_3} of {T}, representing high, medium, and low eigenvalues of {T}, then become a decomposition associated to high, medium, and low Fourier coefficients of {A}.

Remark 3 The use of spectral theory here is parallel to the use of Fourier analysis to establish results such as Roth’s theorem on arithmetic progressions of length three. In analogy with this, one could view hypergraph regularity as being a sort of “higher order spectral theory”, although this spectral perspective is not as convenient as it is in the graph case.

I’ve just uploaded to the arXiv my joint paper with Vitaly Bergelson, “Multiple recurrence in quasirandom groups“, which is submitted to Geom. Func. Anal.. This paper builds upon a paper of Gowers in which he introduced the concept of a quasirandom group, and established some mixing (or recurrence) properties of such groups. A {D}-quasirandom group is a finite group with no non-trivial unitary representations of dimension at most {D}. We will informally refer to a “quasirandom group” as a {D}-quasirandom group with the quasirandomness parameter {D} large (more formally, one can work with a sequence of {D_n}-quasirandom groups with {D_n} going to infinity). A typical example of a quasirandom group is {SL_2(F_p)} where {p} is a large prime. Quasirandom groups are discussed in depth in this blog post. One of the key properties of quasirandom groups established in Gowers’ paper is the following “weak mixing” property: if {A, B} are subsets of {G}, then for “almost all” {g \in G}, one has

\displaystyle  \mu( A \cap gB ) \approx \mu(A) \mu(B) \ \ \ \ \ (1)

where {\mu(A) := |A|/|G|} denotes the density of {A} in {G}. Here, we use {x \approx y} to informally represent an estimate of the form {x=y+o(1)} (where {o(1)} is a quantity that goes to zero when the quasirandomness parameter {D} goes to infinity), and “almost all {g \in G}” denotes “for all {g} in a subset of {G} of density {1-o(1)}“. As a corollary, if {A,B,C} have positive density in {G} (by which we mean that {\mu(A)} is bounded away from zero, uniformly in the quasirandomness parameter {D}, and similarly for {B,C}), then (if the quasirandomness parameter {D} is sufficiently large) we can find elements {g, x \in G} such that {g \in A}, {x \in B}, {gx \in C}. In fact we can find approximately {\mu(A)\mu(B)\mu(C) |G|^2} such pairs {(g,x)}. To put it another way: if we choose {g,x} uniformly and independently at random from {G}, then the events {g \in A}, {x \in B}, {gx \in C} are approximately independent (thus the random variable {(g,x,gx) \in G^3} resembles a uniformly distributed random variable on {G^3} in some weak sense). One can also express this mixing property in integral form as

\displaystyle  \int_G \int_G f_1(g) f_2(x) f_3(gx)\ d\mu(g) d\mu(x) \approx (\int_G f_1\ d\mu) (\int_G f_2\ d\mu) (\int_G f_3\ d\mu)

for any bounded functions {f_1,f_2,f_3: G \rightarrow {\bf R}}. (Of course, with {G} being finite, one could replace the integrals here by finite averages if desired.) Or in probabilistic language, we have

\displaystyle  \mathop{\bf E} f_1(g) f_2(x) f_3(gx) \approx \mathop{\bf E} f_1(x_1) f_2(x_2) f_3(x_3)

where {g, x, x_1, x_2, x_3} are drawn uniformly and independently at random from {G}.

As observed in Gowers’ paper, one can iterate this observation to find “parallelopipeds” of any given dimension in dense subsets of {G}. For instance, applying (1) with {A,B,C} replaced by {A \cap hB}, {C \cap hD}, and {E \cap hF} one can assert (after some relabeling) that for {g,h,x} chosen uniformly and independently at random from {G}, the events {g \in A}, {h \in B}, {gh \in C}, {x \in D}, {gx \in E}, {hx \in F}, {ghx \in H} are approximately independent whenever {A,B,C,D,E,F,H} are dense subsets of {G}; thus the tuple {(g,h,gh,x,gh,hx,ghx)} resebles a uniformly distributed random variable in {G^7} in some weak sense.

However, there are other tuples for which the above iteration argument does not seem to apply. One of the simplest tuples in this vein is the tuple {(g, x, xg, gx)} in {G^4}, when {g, x} are drawn uniformly at random from a quasirandom group {G}. Here, one does not expect the tuple to behave as if it were uniformly distributed in {G^4}, because there is an obvious constraint connecting the last two components {gx, xg} of this tuple: they must lie in the same conjugacy class! In particular, if {A} is a subset of {G} that is the union of conjugacy classes, then the events {gx \in A}, {xg \in A} are perfectly correlated, so that {\mu( gx \in A, xg \in A)} is equal to {\mu(A)} rather than {\mu(A)^2}. Our main result, though, is that in a quasirandom group, this is (approximately) the only constraint on the tuple. More precisely, we have

Theorem 1 Let {G} be a {D}-quasirandom group, and let {g, x} be drawn uniformly at random from {G}. Then for any {f_1,f_2,f_3,f_4: G \rightarrow [-1,1]}, we have

\displaystyle  \mathop{\bf E} f_1(g) f_2(x) f_3(gx) f_4(xg) = \mathop{\bf E} f_1(x_1) f_2(x_2) f_3(x_3) f_4(x_4) + o(1)

where {o(1)} goes to zero as {D \rightarrow \infty}, {x_1,x_2,x_3} are drawn uniformly and independently at random from {G}, and {x_4} is drawn uniformly at random from the conjugates of {x_3} for each fixed choice of {x_1,x_2,x_3}.

This is the probabilistic formulation of the above theorem; one can also phrase the theorem in other formulations (such as an integral formulation), and this is detailed in the paper. This theorem leads to a number of recurrence results; for instance, as a corollary of this result, we have

\displaystyle  \mu(A) \mu(B)^2 - o(1) \leq \mu( A \cap gB \cap Bg ) \leq \mu(A) \mu(B) + o(1)

for almost all {g \in G}, and any dense subsets {A, B} of {G}; the lower and upper bounds are sharp, with the lower bound being attained when {B} is randomly distributed, and the upper bound when {B} is conjugation-invariant.

To me, the more interesting thing here is not the result itself, but how it is proven. Vitaly and I were not able to find a purely finitary way to establish this mixing theorem. Instead, we had to first use the machinery of ultraproducts (as discussed in this previous post) to convert the finitary statement about a quasirandom group to an infinitary statement about a type of infinite group which we call an ultra quasirandom group (basically, an ultraproduct of increasingly quasirandom finite groups). This is analogous to how the Furstenberg correspondence principle is used to convert a finitary combinatorial problem into an infinitary ergodic theory problem.

Ultra quasirandom groups come equipped with a finite, countably additive measure known as Loeb measure {\mu_G}, which is very analogous to the Haar measure of a compact group, except that in the case of ultra quasirandom groups one does not quite have a topological structure that would give compactness. Instead, one has a slightly weaker structure known as a {\sigma}-topology, which is like a topology except that open sets are only closed under countable unions rather than arbitrary ones. There are some interesting measure-theoretic and topological issues regarding the distinction between topologies and {\sigma}-topologies (and between Haar measure and Loeb measure), but for this post it is perhaps best to gloss over these issues and pretend that ultra quasirandom groups {G} come with a Haar measure. One can then recast Theorem 1 as a mixing theorem for the left and right actions of the ultra approximate group {G} on itself, which roughly speaking is the assertion that

\displaystyle  \int_G f_1(x) L_g f_2(x) L_g R_g f_3(x)\ d\mu_G(x) \approx 0 \ \ \ \ \ (2)

for “almost all” {g \in G}, if {f_1, f_2, f_3} are bounded measurable functions on {G}, with {f_3} having zero mean on all conjugacy classes of {G}, where {L_g, R_g} are the left and right translation operators

\displaystyle  L_g f(x) := f(g^{-1} x); \quad R_g f(x) := f(xg).

To establish this mixing theorem, we use the machinery of idempotent ultrafilters, which is a particularly useful tool for understanding the ergodic theory of actions of countable groups {G} that need not be amenable; in the non-amenable setting the classical ergodic averages do not make much sense, but ultrafilter-based averages are still available. To oversimplify substantially, the idempotent ultrafilter arguments let one establish mixing estimates of the form (2) for “many” elements {g} of an infinite-dimensional parallelopiped known as an IP system (provided that the actions {L_g,R_g} of this IP system obey some technical mixing hypotheses, but let’s ignore that for sake of this discussion). The claim then follows by using the quasirandomness hypothesis to show that if the estimate (2) failed for a large set of {g \in G}, then this large set would then contain an IP system, contradicting the previous claim.

Idempotent ultrafilters are an extremely infinitary type of mathematical object (one has to use Zorn’s lemma no fewer than three times just to construct one of these objects!). So it is quite remarkable that they can be used to establish a finitary theorem such as Theorem 1, though as is often the case with such infinitary arguments, one gets absolutely no quantitative control whatsoever on the error terms {o(1)} appearing in that theorem. (It is also mildly amusing to note that our arguments involve the use of ultrafilters in two completely different ways: firstly in order to set up the ultraproduct that converts the finitary mixing problem to an infinitary one, and secondly to solve the infinitary mixing problem. Despite some superficial similarities, there appear to be no substantial commonalities between these two usages of ultrafilters.) There is already a fair amount of literature on using idempotent ultrafilter methods in infinitary ergodic theory, and perhaps by further development of ultraproduct correspondence principles, one can use such methods to obtain further finitary consequences (although the state of the art for idempotent ultrafilter ergodic theory has not advanced much beyond the analysis of two commuting shifts {L_g, R_g} currently, which is the main reason why our arguments only handle the pattern {(g,x,xg,gx)} and not more sophisticated patterns).

We also have some miscellaneous other results in the paper. It turns out that by using the triangle removal lemma from graph theory, one can obtain a recurrence result that asserts that whenever {A} is a dense subset of a finite group {G} (not necessarily quasirandom), then there are {\gg |G|^2} pairs {(x,g)} such that {x, gx, xg} all lie in {A}. Using a hypergraph generalisation of the triangle removal lemma known as the hypergraph removal lemma, one can obtain more complicated versions of this statement; for instance, if {A} is a dense subset of {G^2}, then one can find {\gg |G|^2} triples {(x,y,g)} such that {(x,y), (gx, y), (gx, gy), (gxg^{-1}, gyg^{-1})} all lie in {A}. But the method is tailored to the specific types of patterns given here, and we do not have a general method for obtaining recurrence or mixing properties for arbitrary patterns of words in some finite alphabet such as {g,x,y}.

We also give some properties of a model example of an ultra quasirandom group, namely the ultraproduct {SL_2(F)} of {SL_2(F_{p_n})} where {p_n} is a sequence of primes going off to infinity. Thanks to the substantial recent progress (by Helfgott, Bourgain, Gamburd, Breuillard, and others) on understanding the expansion properties of the finite groups {SL_2(F_{p_n})}, we have a fair amount of knowledge on the ultraproduct {SL_2(F)} as well; for instance any two elements of {SL_2(F)} will almost surely generate a spectral gap. We don’t have any direct application of this particular ultra quasirandom group, but it might be interesting to study it further.

I’ve just uploaded to the arXiv my paper “Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets“, submitted to Contrib. Disc. Math. The motivation of this paper is to understand a certain polynomial variant of the sum-product phenomenon in finite fields. This phenomenon asserts that if {A} is a non-empty subset of a finite field {F}, then either the sumset {A+A := \{a+b: a,b \in A\}} or product set {A \cdot A := \{ab: a,b \in A \}} will be significantly larger than {A}, unless {A} is close to a subfield of {F} (or to {\{1\}}). In particular, in the regime when {A} is large, say {|F|^{1-c} < |A| \leq |F|}, one expects an expansion bound of the form

\displaystyle  |A+A| + |A \cdot A| \gg (|F|/|A|)^{c'} |A| \ \ \ \ \ (1)

for some absolute constants {c, c' > 0}. Results of this type are known; for instance, Hart, Iosevich, and Solymosi obtained precisely this bound for {(c,c')=(3/10,1/3)} (in the case when {|F|} is prime), which was then improved by Garaev to {(c,c')=(1/3,1/2)}.

We have focused here on the case when {A} is a large subset of {F}, but sum-product estimates are also extremely interesting in the opposite regime in which {A} is allowed to be small (see for instance the papers of Katz-Shen and Li and of Garaev for some recent work in this case, building on some older papers of Bourgain, Katz and myself and of Bourgain, Glibichuk, and Konyagin). However, the techniques used in these two regimes are rather different. For large subsets of {F}, it is often profitable to use techniques such as the Fourier transform or the Cauchy-Schwarz inequality to “complete” a sum over a large set (such as {A}) into a set over the entire field {F}, and then to use identities concerning complete sums (such as the Weil bound on complete exponential sums over a finite field). For small subsets of {F}, such techniques are usually quite inefficient, and one has to proceed by somewhat different combinatorial methods which do not try to exploit the ambient field {F}. But my paper focuses exclusively on the large {A} regime, and unfortunately does not directly say much (except through reasoning by analogy) about the small {A} case.

Note that it is necessary to have both {A+A} and {A \cdot A} appear on the left-hand side of (1). Indeed, if one just has the sumset {A+A}, then one can set {A} to be a long arithmetic progression to give counterexamples to (1). Similarly, if one just has a product set {A \cdot A}, then one can set {A} to be a long geometric progression. The sum-product phenomenon can then be viewed that it is not possible to simultaneously behave like a long arithmetic progression and a long geometric progression, unless one is already very close to behaving like a subfield.

Now we consider a polynomial variant of the sum-product phenomenon, where we consider a polynomial image

\displaystyle  P(A,A) := \{ P(a,b): a,b \in A \}

of a set {A \subset F} with respect to a polynomial {P: F \times F \rightarrow F}; we can also consider the asymmetric setting of the image

\displaystyle  P(A,B) := \{ P(a,b): a \in A,b \in B \}

of two subsets {A,B \subset F}. The regime we will be interested is the one where the field {F} is large, and the subsets {A, B} of {F} are also large, but the polynomial {P} has bounded degree. Actually, for technical reasons it will not be enough for us to assume that {F} has large cardinality; we will also need to assume that {F} has large characteristic. (The two concepts are synonymous for fields of prime order, but not in general; for instance, the field with {2^n} elements becomes large as {n \rightarrow \infty} while the characteristic remains fixed at {2}, and is thus not going to be covered by the results in this paper.)

In this paper of Vu, it was shown that one could replace {A \cdot A} with {P(A,A)} in (1), thus obtaining a bound of the form

\displaystyle  |A+A| + |P(A,A)| \gg (|F|/|A|)^{c'} |A|

whenever {|A| \geq |F|^{1-c}} for some absolute constants {c, c' > 0}, unless the polynomial {P} had the degenerate form {P(x,y) = Q(L(x,y))} for some linear function {L: F \times F \rightarrow F} and polynomial {Q: F \rightarrow F}, in which {P(A,A)} behaves too much like {A+A} to get reasonable expansion. In this paper, we focus instead on the question of bounding {P(A,A)} alone. In particular, one can ask to classify the polynomials {P} for which one has the weak expansion property

\displaystyle |P(A,A)| \gg (|F|/|A|)^{c'} |A|

whenever {|A| \geq |F|^{1-c}} for some absolute constants {c, c' > 0}. One can also ask for stronger versions of this expander property, such as the moderate expansion property

\displaystyle |P(A,A)| \gg |F|

whenever {|A| \geq |F|^{1-c}}, or the almost strong expansion property

\displaystyle |P(A,A)| \geq |F| - O( |F|^{1-c'})

whenever {|A| \geq |F|^{1-c}}. (One can consider even stronger expansion properties, such as the strong expansion property {|P(A,A)| \geq |F|-O(1)}, but it was shown by Gyarmati and Sarkozy that this property cannot hold for polynomials of two variables of bounded degree when {|F| \rightarrow \infty}.) One can also consider asymmetric versions of these properties, in which one obtains lower bounds on {|P(A,B)|} rather than {|P(A,A)|}.

The example of a long arithmetic or geometric progression shows that the polynomials {P(x,y) = x+y} or {P(x,y) = xy} cannot be expanders in any of the above senses, and a similar construction also shows that polynomials of the form {P(x,y) = Q(f(x)+f(y))} or {P(x,y) = Q(f(x) f(y))} for some polynomials {Q, f: F \rightarrow F} cannot be expanders. On the other hand, there are a number of results in the literature establishing expansion for various polynomials in two or more variables that are not of this degenerate form (in part because such results are related to incidence geometry questions in finite fields, such as the finite field version of the Erdos distinct distances problem). For instance, Solymosi established weak expansion for polynomials of the form {P(x,y) = f(x)+y} when {f} is a nonlinear polynomial, with generalisations by Hart, Li, and Shen for various polynomials of the form {P(x,y) = f(x) + g(y)} or {P(x,y) = f(x) g(y)}. Further examples of expanding polynomials appear in the work of Shkredov, Iosevich-Rudnev, and Bukh-Tsimerman, as well as the previously mentioned paper of Vu and of Hart-Li-Shen, and these papers in turn cite many further results which are in the spirit of the polynomial expansion bounds discussed here (for instance, dealing with the small {A} regime, or working in other fields such as {{\bf R}} instead of in finite fields {F}). We will not summarise all these results here; they are summarised briefly in my paper, and in more detail in the papers of Hart-Li-Shen and of Bukh-Tsimerman. But we will single out one of the results of Bukh-Tsimerman, which is one of most recent and general of these results, and closest to the results of my own paper. Roughly speaking, in this paper it is shown that a polynomial {P(x,y)} of two variables and bounded degree will be a moderate expander if it is non-composite (in the sense that it does not take the form {P(x,y) = Q(R(x,y))} for some non-linear polynomial {Q} and some polynomial {R}, possibly having coefficients in the algebraic completion of {F}) and is monic on both {x} and {y}, thus it takes the form {P(x,y) = x^d + S(x,y)} for some {d \geq 1} and some polynomial {S} of degree at most {d-1} in {x}, and similarly with the roles of {x} and {y} reversed, unless {P} is of the form {P(x,y) = f(x) + g(y)} or {P(x,y) = f(x) g(y)} (in which case the expansion theory is covered to a large extent by the previous work of Hart, Li, and Shen).

Our first main result improves upon the Bukh-Tsimerman result by strengthening the notion of expansion and removing the non-composite and monic hypotheses, but imposes a condition of large characteristic. I’ll state the result here slightly informally as follows:

Theorem 1 (Criterion for moderate expansion) Let {P: F \times F \rightarrow F} be a polynomial of bounded degree over a finite field {F} of sufficiently large characteristic, and suppose that {P} is not of the form {P(x,y) = Q(f(x)+g(y))} or {P(x,y) = Q(f(x)g(y))} for some polynomials {Q,f,g: F \rightarrow F}. Then one has the (asymmetric) moderate expansion property

\displaystyle  |P(A,B)| \gg |F|

whenever {|A| |B| \ggg |F|^{2-1/8}}.

This is basically a sharp necessary and sufficient condition for asymmetric expansion moderate for polynomials of two variables. In the paper, analogous sufficient conditions for weak or almost strong expansion are also given, although these are not quite as satisfactory (particularly the conditions for almost strong expansion, which include a somewhat complicated algebraic condition which is not easy to check, and which I would like to simplify further, but was unable to).

The argument here resembles the Bukh-Tsimerman argument in many ways. One can view the result as an assertion about the expansion properties of the graph {\{ (a,b,P(a,b)): a,b \in F \}}, which can essentially be thought of as a somewhat sparse three-uniform hypergraph on {F}. Being sparse, it is difficult to directly apply techniques from dense graph or hypergraph theory for this situation; however, after a few applications of the Cauchy-Schwarz inequality, it turns out (as observed by Bukh and Tsimerman) that one can essentially convert the problem to one about the expansion properties of the set

\displaystyle  \{ (P(a,c), P(b,c), P(a,d), P(b,d)): a,b,c,d \in F \} \ \ \ \ \ (2)

(actually, one should view this as a multiset, but let us ignore this technicality) which one expects to be a dense set in {F^4}, except in the case when the associated algebraic variety

\displaystyle  \{ (P(a,c), P(b,c), P(a,d), P(b,d)): a,b,c,d \in \overline{F} \}

fails to be Zariski dense, but it turns out that in this case one can use some differential geometry and Riemann surface arguments (after first invoking the Lefschetz principle and the high characteristic hypothesis to work over the complex numbers instead over a finite field) to show that {P} is of the form {Q(f(x)+g(y))} or {Q(f(x)g(y))}. This reduction is related to the classical fact that the only one-dimensional algebraic groups over the complex numbers are the additive group {({\bf C},+)}, the multiplicative group {({\bf C} \backslash \{0\},\times)}, or the elliptic curves (but the latter have a group law given by rational functions rather than polynomials, and so ultimately end up being eliminated from consideration, though they would play an important role if one wanted to study the expansion properties of rational functions).

It remains to understand the structure of the set (2) is. To understand dense graphs or hypergraphs, one of the standard tools of choice is the Szemerédi regularity lemma, which carves up such graphs into a bounded number of cells, with the graph behaving pseudorandomly on most pairs of cells. However, the bounds in this lemma are notoriously poor (the regularity obtained is an inverse tower exponential function of the number of cells), and this makes this lemma unsuitable for the type of expansion properties we seek (in which we want to deal with sets {A} which have a polynomial sparsity, e.g. {|A| \sim |F|^{1-c}}). Fortunately, in the case of sets such as (2) which are definable over the language of rings, it turns out that a much stronger regularity lemma is available, which I call the “algebraic regularity lemma”. I’ll state it (again, slightly informally) in the context of graphs as follows:

Lemma 2 (Algebraic regularity lemma) Let {F} be a finite field of large characteristic, and let {V, W} be definable sets over {F} of bounded complexity (i.e. {V, W} are subsets of {F^n}, {F^m} for some bounded {n,m} that can be described by some first-order predicate in the language of rings of bounded length and involving boundedly many constants). Let {E} be a definable subset of {V \times W}, again of bounded complexity (one can view {E} as a bipartite graph connecting {V} and {W}). Then one can partition {V, W} into a bounded number of cells {V_1,\ldots,V_a}, {W_1,\ldots,W_b}, still definable with bounded complexity, such that for all pairs {i =1,\ldots a}, {j=1,\ldots,b}, one has the regularity property

\displaystyle  |E \cap (A \times B)| = d_{ij} |A| |B| + O( |F|^{-1/4} |V| |W| )

for all {A \subset V_i, B \subset W_i}, where {d_{ij} := \frac{|E \cap (V_i \times W_j)|}{|V_i| |W_j|}} is the density of {E} in {V_i \times W_j}.

This lemma resembles the Szemerédi regularity lemma, but regularises all pairs of cells (not just most pairs), and the regularity is of polynomial strength in {|F|}, rather than inverse tower exponential in the number of cells. Also, the cells are not arbitrary subsets of {V,W}, but are themselves definable with bounded complexity, which turns out to be crucial for applications. I am optimistic that this lemma will be useful not just for studying expanding polynomials, but for many other combinatorial questions involving dense subsets of definable sets over finite fields.

The above lemma is stated for graphs {E \subset V \times W}, but one can iterate it to obtain an analogous regularisation of hypergraphs {E \subset V_1 \times \ldots \times V_k} for any bounded {k} (for application to (2), we need {k=4}). This hypergraph regularity lemma, by the way, is not analogous to the strong hypergraph regularity lemmas of Rodl et al. and Gowers developed in the last six or so years, but closer in spirit to the older (but weaker) hypergraph regularity lemma of Chung which gives the same “order {1}” regularity that the graph regularity lemma gives, rather than higher order regularity.

One feature of the proof of Lemma 2 which I found striking was the need to use some fairly high powered technology from algebraic geometry, and in particular the Lang-Weil bound on counting points in varieties over a finite field (discussed in this previous blog post), and also the theory of the etale fundamental group. Let me try to briefly explain why this is the case. A model example of a definable set of bounded complexity {E} is a set {E \subset F^n \times F^m} of the form

\displaystyle  E = \{ (x,y) \in F^n \times F^m: \exists t \in F; P(x,y,t)=0 \}

for some polynomial {P: F^n \times F^m \times F \rightarrow F}. (Actually, it turns out that one can essentially write all definable sets as an intersection of sets of this form; see this previous blog post for more discussion.) To regularise the set {E}, it is convenient to square the adjacency matrix, which soon leads to the study of counting functions such as

\displaystyle  \mu(x,x') := | \{ (y,t,t') \in F^m \times F \times F: P(x,y,t) = P(x',y,t') = 0 \}|.

If one can show that this function {\mu} is “approximately finite rank” in the sense that (modulo lower order errors, of size {O(|F|^{-1/2})} smaller than the main term), this quantity depends only on a bounded number of bits of information about {x} and a bounded number of bits of information about {x'}, then a little bit of linear algebra will then give the required regularity result.

One can recognise {\mu(x,x')} as counting {F}-points of a certain algebraic variety

\displaystyle  V_{x,x'} := \{ (y,t,t') \in \overline{F}^m \times \overline{F} \times \overline{F}: P(x,y,t) = P(x',y,t') = 0 \}.

The Lang-Weil bound (discussed in this previous post) provides a formula for this count, in terms of the number {c(x,x')} of geometrically irreducible components of {V_{x,x'}} that are defined over {F} (or equivalently, are invariant with respect to the Frobenius endomorphism associated to {F}). So the problem boils down to ensuring that this quantity {c(x,x')} is “generically bounded rank”, in the sense that for generic {x,x'}, its value depends only on a bounded number of bits of {x} and a bounded number of bits of {x'}.

Here is where the étale fundamental group comes in. One can view {V_{x,x'}} as a fibre product {V_x \times_{\overline{F}^m} V_{x'}} of the varieties

\displaystyle  V_x := \{ (y,t) \in \overline{F}^m \times \overline{F}: P(x,y,t) = 0 \}

and

\displaystyle  V_{x'} := \{ (y,t) \in \overline{F}^m \times \overline{F}: P(x',y,t) = 0 \}

over {\overline{F}^m}. If one is in sufficiently high characteristic (or even better, in zero characteristic, which one can reduce to by an ultraproduct (or nonstandard analysis) construction, similar to that discussed in this previous post), the varieties {V_x,V_{x'}} are generically finite étale covers of {\overline{F}^m}, and the fibre product {V_x \times_{\overline{F}^m} V_{x'}} is then also generically a finite étale cover. One can count the components of a finite étale cover of a connected variety by counting the number of orbits of the étale fundamental group acting on a fibre of that variety (much as the number of components of a cover of a connected manifold is the number of orbits of the topological fundamental group acting on that fibre). So if one understands the étale fundamental group of a certain generic subset of {\overline{F}^m} (formed by intersecting together an {x}-dependent generic subset of {\overline{F}^m} with an {x'}-dependent generic subset), this in principle controls {c(x,x')}. It turns out that one can decouple the {x} and {x'} dependence of this fundamental group by using an étale version of the van Kampen theorem for the fundamental group, which I discussed in this previous blog post. With this fact (and another deep fact about the étale fundamental group in zero characteristic, namely that it is topologically finitely generated), one can obtain the desired generic bounded rank property of {c(x,x')}, which gives the regularity lemma.

In order to expedite the deployment of all this algebraic geometry (as well as some Riemann surface theory), it is convenient to use the formalism of nonstandard analysis (or the ultraproduct construction), which among other things can convert quantitative, finitary problems in large characteristic into equivalent qualitative, infinitary problems in zero characteristic (in the spirit of this blog post). This allows one to use several tools from those fields as “black boxes”; not just the theory of étale fundamental groups (which are considerably simpler and more favorable in characteristic zero than they are in positive characteristic), but also some results limiting the morphisms between compact Riemann surfaces of high genus (such as the de Franchis theorem, the Riemann-Hurwitz formula, or the fact that all morphisms between elliptic curves are essentially group homomorphisms), which would be quite unwieldy to utilise if one did not first pass to the zero characteristic case (and thence to the complex case) via the ultraproduct construction (followed by the Lefschetz principle).

I found this project to be particularly educational for me, as it forced me to wander outside of my usual range by quite a bit in order to pick up the tools from algebraic geometry and Riemann surfaces that I needed (in particular, I read through several chapters of EGA and SGA for the first time). This did however put me in the slightly unnerving position of having to use results (such as the Riemann existence theorem) whose proofs I have not fully verified for myself, but which are easy to find in the literature, and widely accepted in the field. I suppose this type of dependence on results in the literature is more common in the more structured fields of mathematics than it is in analysis, which by its nature has fewer reusable black boxes, and so key tools often need to be rederived and modified for each new application. (This distinction is discussed further in this article of Gowers.)

One of the first non-trivial theorems one encounters in classical algebraic geometry is Bézout’s theorem, which we will phrase as follows:

Theorem 1 (Bézout’s theorem) Let {k} be a field, and let {P, Q \in k[x,y]} be non-zero polynomials in two variables {x,y} with no common factor. Then the two curves {\{ (x,y) \in k^2: P(x,y) = 0\}} and {\{ (x,y) \in k^2: Q(x,y) = 0\}} have no common components, and intersect in at most {\hbox{deg}(P) \hbox{deg}(Q)} points.

This theorem can be proven by a number of means, for instance by using the classical tool of resultants. It has many strengthenings, generalisations, and variants; see for instance this previous blog post on Bézout’s inequality. Bézout’s theorem asserts a fundamental algebraic dichotomy, of importance in combinatorial incidence geometry: any two algebraic curves either share a common component, or else have a bounded finite intersection; there is no intermediate case in which the intersection is unbounded in cardinality, but falls short of a common component. This dichotomy is closely related to the integrality gap in algebraic dimension: an algebraic set can have an integer dimension such as {0} or {1}, but cannot attain any intermediate dimensions such as {1/2}. This stands in marked contrast to sets of analytic, combinatorial, or probabilistic origin, whose “dimension” is typically not necessarily constrained to be an integer.

Bézout’s inequality tells us, roughly speaking, that the intersection of a curve of degree {D_1} and a curve of degree {D_2} forms a set of at most {D_1 D_2} points. One can consider the converse question: given a set {S} of {N} points in the plane {k^2}, can one find two curves of degrees {D_1,D_2} with {D_1 D_2 = O(N)} and no common components, whose intersection contains these points?

A model example that supports the possibility of such a converse is a grid {S = A \times B} that is a Cartesian product of two finite subsets {A, B} of {k} with {|A| |B| = N}. In this case, one can take one curve to be the union of {|A|} vertical lines, and the other curve to be the union of {|B|} horizontal lines, to obtain the required decomposition. Thus, if the proposed converse to Bézout’s inequality held, it would assert that any set of {N} points was essentially behaving like a “nonlinear grid” of size {N}.

Unfortunately, the naive converse to Bézout’s theorem is false. A counterexample can be given by considering a set {S = S_1 \cup S_2} of {2N} points for some large perfect square {N}, where {P_1} is a {\sqrt{N}} by {\sqrt{N}} grid of the form described above, and {S_2} consists of {N} points on an line {\ell} (e.g. a {1 \times N} or {N \times 1} grid). Each of the two component sets {S_1, S_2} can be written as the intersection between two curves whose degrees multiply up to {N}; in the case of {S_1}, we can take the two families of parallel lines (viewed as reducible curves of degree {\sqrt{N}}) as the curves, and in the case of {S_2}, one can take {\ell} as one curve, and the graph of a degree {N} polynomial on {\ell} vanishing on {S_2} for the second curve. But, if {N} is large enough, one cannot cover {S} by the intersection of a single pair {\gamma_1, \gamma_2} of curves with no common components whose degrees {D_1,D_2} multiply up to {D_1 D_2 = O(N)}. Indeed, if this were the case, then without loss of generality we may assume that {D_1 \leq D_2}, so that {D_1 = O(\sqrt{N})}. By Bézout’s theorem, {\gamma_1} either contains {\ell}, or intersects {\ell} in at most {O(D_1) = O(\sqrt{N})} points. Thus, in order for {\gamma_1} to capture all of {S}, it must contain {\ell}, which forces {\gamma_2} to not contain {\ell}. But {\gamma_2} has to intersect {\ell} in {N} points, so by Bézout’s theorem again we have {D_2 \geq N}, thus {D_1 = O(1)}. But then (by more applications of Bézout’s theorem) {\gamma_1} can only capture {O(\sqrt{N})} of the {N} points of {S_1}, a contradiction.

But the above counterexample suggests that even if an arbitrary set of {N} (or {2N}) points cannot be covered by the single intersection of a pair of curves with degree multiplying up to {O(N)}, one may be able to cover such a set by a small number of such intersections. The purpose of this post is to record the simple observation that this is, indeed, the case:

Theorem 2 (Partial converse to Bézout’s theorem) Let {k} be a field, and let {S} be a set of {N} points in {k} for some {N > 1}. Then one can find {m = O(\log N)} and pairs {P_i,Q_i \in k[x,y]} of coprime non-zero polynomials for {i=1,\ldots,m} such that

\displaystyle  S \subset \bigcup_{i=1}^m \{ (x,y) \in k^2: P_i(x,y) = Q_i(x,y) = 0 \} \ \ \ \ \ (1)

and

\displaystyle  \sum_{i=1}^m \hbox{deg}(P_i) \hbox{deg}(Q_i) = O( N ). \ \ \ \ \ (2)

Informally, every finite set in the plane is (a dense subset of) the union of logarithmically many nonlinear grids. The presence of the logarithm is necessary, as can be seen by modifying the {P_1 \cup P_2} example to be the union of logarithmically many Cartesian products of distinct dimensions, rather than just a pair of such products.

Unfortunately I do not know of any application of this converse, but I thought it was cute anyways. The proof is given below the fold.

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