arithmetic-combinatorics &laquo; WordPress.com Tag Feed http://en.wordpress.com/tag/arithmetic-combinatorics/ Feed of posts on WordPress.com tagged "arithmetic-combinatorics" Sat, 25 May 2013 20:45:22 +0000 http://en.wordpress.com/tags/ en <![CDATA[Rectification and the Lefschetz principle]]> http://terrytao.wordpress.com/2013/03/14/rectification-and-the-lefschetz-principle/ Thu, 14 Mar 2013 21:17:05 +0000 Terence Tao http://terrytao.wordpress.com/2013/03/14/rectification-and-the-lefschetz-principle/ 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}}&fg=000000 or the complex numbers {{\bf C}}&fg=000000. 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}&fg=000000 be a subset of the additive group {{\bf Z}/p{\bf Z}}&fg=000000 for some prime {p}&fg=000000, and let {s \geq 1}&fg=000000 be an integer. Suppose that {|A| \leq \log_{2s} p}&fg=000000. Then there exists a map {\phi: A \rightarrow A'}&fg=000000 into a subset {A'}&fg=000000 of the integers which is a Freiman isomorphism of order {s}&fg=000000 in the sense that for any {x_1,\ldots,x_s,y_1,\ldots,y_s \in A}&fg=000000, one has

\displaystyle x_1+\ldots+x_s = y_1+\ldots+y_s&fg=000000

if and only if

\displaystyle \phi(x_1)+\ldots+\phi(x_s) = \phi(y_1)+\ldots+\phi(y_s).&fg=000000

Furthermore {\phi}&fg=000000 is a right-inverse of the obvious projection homomorphism from {{\bf Z}}&fg=000000 to {{\bf Z}/p{\bf Z}}&fg=000000.

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}&fg=000000), 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}&fg=000000 of {A}&fg=000000 that is contained in a small neighbourhood of the origin in {{\bf Z}/p{\bf Z}}&fg=000000, at which point the rectification map {\phi}&fg=000000 can be constructed by hand.

Very recently, Codrut Grosu obtained an arithmetic analogue of the above theorem, in which the rectification map {\phi}&fg=000000 preserves both additive and multiplicative structure:

Theorem 2 (Arithmetic rectification) Let {A}&fg=000000 be a subset of the finite field {{\bf F}_p}&fg=000000 for some prime {p \geq 3}&fg=000000, and let {s \geq 1}&fg=000000 be an integer. Suppose that {|A| < \log_2 \log_{2s} \log_{2s^2} p - 1}&fg=000000. Then there exists a map {\phi: A \rightarrow A'}&fg=000000 into a subset {A'}&fg=000000 of the complex numbers which is a Freiman field isomorphism of order {s}&fg=000000 in the sense that for any {x_1,\ldots,x_n \in A}&fg=000000 and any polynomial {P(x_1,\ldots,x_n)}&fg=000000 of degree at most {s}&fg=000000 and integer coefficients of magnitude summing to at most {s}&fg=000000, one has

\displaystyle P(x_1,\ldots,x_n)=0&fg=000000

if and only if

\displaystyle P(\phi(x_1),\ldots,\phi(x_n))=0.&fg=000000

Note that it is necessary to use an algebraically closed field such as {{\bf C}}&fg=000000 for this theorem, in contrast to the integers used in Proposition 1, as {{\bf F}_p}&fg=000000 can contain objects such as square roots of {-1}&fg=000000 which can only map to {\pm i}&fg=000000 in the complex numbers (once {s}&fg=000000 is at least {2}&fg=000000).

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}}&fg=000000 to analogous results regarding sufficiently small subsets of {{\bf F}_p}&fg=000000; 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}}&fg=000000 (or more generally, a characteristic zero integral domain) in a Freiman field-isomorphic fashion into finite subsets of {{\bf F}_p}&fg=000000 for arbitrarily large primes {p}&fg=000000, 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}&fg=000000. Then if {{\bf F}}&fg=000000 is a field of characteristic at least {C_{s,n}}&fg=000000 for some {C_{s,n}}&fg=000000 depending on {s,n}&fg=000000, and {A}&fg=000000 is a subset of {{\bf F}}&fg=000000 of cardinality {n}&fg=000000, then there exists a map {\phi: A \rightarrow A'}&fg=000000 into a subset {A'}&fg=000000 of the complex numbers which is a Freiman field isomorphism of order {s}&fg=000000.

Our arguments will not provide any effective bound on the quantity {C_{s,n}}&fg=000000 (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}&fg=000000 be a field of characteristic zero that is finitely generated over the rationals. Then there is an isomorphism {\phi: k \rightarrow \phi(k)}&fg=000000 from {k}&fg=000000 to a subfield {\phi(k)}&fg=000000 of {{\bf C}}&fg=000000.

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}&fg=000000 is finitely generated over {{\bf Q}}&fg=000000, it has finite transcendence degree, thus one can find algebraically independent elements {x_1,\ldots,x_m}&fg=000000 of {k}&fg=000000 over {{\bf Q}}&fg=000000 such that {k}&fg=000000 is a finite extension of {{\bf Q}(x_1,\ldots,x_m)}&fg=000000, and in particular by the primitive element theorem {k}&fg=000000 is generated by {{\bf Q}(x_1,\ldots,x_m)}&fg=000000 and an element {\alpha}&fg=000000 which is algebraic over {{\bf Q}(x_1,\ldots,x_m)}&fg=000000. (Here we use the fact that characteristic zero fields are separable.) If we then define {\phi}&fg=000000 by first mapping {x_1,\ldots,x_m}&fg=000000 to generic (and thus algebraically independent) complex numbers {z_1,\ldots,z_m}&fg=000000, and then setting {\phi(\alpha)}&fg=000000 to be a complex root of of the minimal polynomial for {\alpha}&fg=000000 over {{\bf Q}(x_1,\ldots,x_m)}&fg=000000 after replacing each {x_i}&fg=000000 with the complex number {z_i}&fg=000000, we obtain a field isomorphism {\phi: k \rightarrow \phi(k)}&fg=000000 with the required properties.

Now we give the latter proof. Let {x_1,\ldots,x_m}&fg=000000 be elements of {k}&fg=000000 that generate that field over {{\bf Q}}&fg=000000, but which are not necessarily algebraically independent. Our task is then equivalent to that of finding complex numbers {z_1,\ldots,z_m}&fg=000000 with the property that, for any polynomial {P(x_1,\ldots,x_m)}&fg=000000 with rational coefficients, one has

\displaystyle P(x_1,\ldots,x_m) = 0&fg=000000

if and only if

\displaystyle P(z_1,\ldots,z_m) = 0.&fg=000000

Let {{\mathcal P}}&fg=000000 be the collection of all polynomials {P}&fg=000000 with rational coefficients with {P(x_1,\ldots,x_m)=0}&fg=000000, and {{\mathcal Q}}&fg=000000 be the collection of all polynomials {P}&fg=000000 with rational coefficients with {P(x_1,\ldots,x_m) \neq 0}&fg=000000. 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} \}&fg=000000

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}&fg=000000 could be covered by the algebraic sets {\{ (z_1,\ldots,z_m) \in {\bf C}^m: Q(z_1,\ldots,z_m) = 0 \}}&fg=000000 for {Q \in {\mathcal Q}}&fg=000000. By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over {{\bf C}}&fg=000000 cannot be covered by countably many varieties of smaller dimension, we conclude that {S}&fg=000000 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 \}&fg=000000

for some {Q_1,\ldots,Q_n \in {\bf C}^m}&fg=000000. 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&fg=000000

for some natural numbers {l,r \geq 1}&fg=000000, polynomials {P_1,\ldots,P_r \in {\mathcal P}}&fg=000000, and polynomials {R_1,\ldots,R_r}&fg=000000 with coefficients in {\overline{{\bf Q}}}&fg=000000. In particular, this identity also holds in the algebraic closure {\overline{k}}&fg=000000 of {k}&fg=000000. Evaluating this identity at {(x_1,\ldots,x_m)}&fg=000000 we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows. \Box&fg=000000

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}&fg=000000, a sequence of finite fields {{\bf F}_i}&fg=000000 of characteristic at least {i}&fg=000000, and subsets {A_i=\{a_{i,1},\ldots,a_{i,n}\}}&fg=000000 of {{\bf F}_i}&fg=000000 of cardinality {n}&fg=000000 such that for each {i}&fg=000000, there does not exist a Freiman field isomorphism of order {s}&fg=000000 from {A_i}&fg=000000 to the complex numbers. Now we select a non-principal ultrafilter {\alpha \in \beta {\bf N} \backslash {\bf N}}&fg=000000, and construct the ultraproduct {{\bf F} := \prod_{i \rightarrow \alpha} {\bf F}_i}&fg=000000 of the finite fields {{\bf F}_i}&fg=000000. 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}&fg=000000 goes to infinity as {i \rightarrow \infty}&fg=000000, it is easy to see (using Los’s theorem) that {{\bf F}}&fg=000000 has characteristic zero and can thus be viewed as an extension of the rationals {{\bf Q}}&fg=000000.

Now let {a_j := \lim_{i \rightarrow \alpha} a_{i,j}}&fg=000000 be the ultralimit of the {a_{i,j}}&fg=000000, so that {A := \{a_1,\ldots,a_n\}}&fg=000000 is the ultraproduct of the {A_i}&fg=000000, then {A}&fg=000000 is a subset of {{\bf F}}&fg=000000 of cardinality {n}&fg=000000. In particular, if {k}&fg=000000 is the field generated by {{\bf Q}}&fg=000000 and {A}&fg=000000, then {k}&fg=000000 is a finitely generated extension of the rationals and thus, by Proposition 4 there is an isomorphism {\phi: k \rightarrow \phi(k)}&fg=000000 from {k}&fg=000000 to a subfield {\phi(k)}&fg=000000 of the complex numbers. In particular, {\phi(a_1),\ldots,\phi(a_n)}&fg=000000 are complex numbers, and for any polynomial {P(x_1,\ldots,x_n)}&fg=000000 with integer coefficients, one has

\displaystyle P(a_1,\ldots,a_n) = 0&fg=000000

if and only if

\displaystyle P(\phi(a_1),\ldots,\phi(a_n)) = 0.&fg=000000

By Los’s theorem, we then conclude that for all {i}&fg=000000 sufficiently close to {\alpha}&fg=000000, one has for all polynomials {P(x_1,\ldots,x_n)}&fg=000000 of degree at most {s}&fg=000000 and whose coefficients are integers whose magnitude sums up to {s}&fg=000000, one has

\displaystyle P(a_{i,1},\ldots,a_{i,n}) = 0&fg=000000

if and only if

\displaystyle P(\phi(a_1),\ldots,\phi(a_n)) = 0.&fg=000000

But this gives a Freiman field isomorphism of order {s}&fg=000000 between {A_i}&fg=000000 and {\phi(A)}&fg=000000, contradicting the construction of {A_i}&fg=000000, and Theorem 3 follows.

]]> <![CDATA[Algebraic Combinatorics]]> http://yabm.wordpress.com/2012/04/11/algebraic-combinatorics/ Wed, 11 Apr 2012 19:36:49 +0000 Adam Bohn http://yabm.wordpress.com/2012/04/11/algebraic-combinatorics/ This must be my most prolific blogging period yet!  Having started this blog over two years ago, I have managed to write a grand total of 30 posts.  Pretty pathetic, I think you’ll agree.  But there we go.  Surprisingly though, in spite of this (or perhaps partly because of it – maybe people like blogs which are not constantly bombarding them with posts to read!) I do have quite a few subscribers.   And the other day I discovered that this site has a page rank of 4 (if you don’t know about PageRank already I wrote about in in this post), which means it must at least been linked to a few times (a bit like the blog version of a citation I suppose).  So there must be some point to it all, and I shall continue for the time-being.

Out of interest, I looked up the PageRank of Facebook, and discovered it is 9.  In fact not even google.com itself has the full 10 out of 10!  I suppose it must be impossible to achieve.  Unfortunately my coffee website, the popularity of which I care – in a way – much more about,  is still floundering with a page rank of 2 (although it has actually recently just made it into the second page of a google search on “Vietnamese coffee”).  Better than I had assumed at least – I stopped advertising with google some time ago and my sales dropped drastically (although profit remained about even – that’s how much pay-per-click advertising costs) and had a rather paranoid suspicion that Google were trying to lure me back into their fold by lowering my ranking…

Anyway, my posts have begun to get a bit autobiographical, which I had originally been intent on avoiding…the last thing anyone wants is to read a detailed description ofwhat someone had for breakfast that day (this is mainly why I avoid Twitter).  So today I am actually going to talk about mathematics, if you can believe that.  Algebraic Combinatorics, to be precise (I’m going to capitalise fields of mathematics to make this whole post easier to read).   But as a way of getting there, permit me one more mention of myself!

The other day I mentioned that I had passed what was quite a mathematical milestone for me – a conference that I really enjoyed.  Well, today I have another small one.  I have found out I’ve been accepted to attend a summer schoool on Algebraic and Enumerative Combinatorics in Portugal.  I don’t think this is an especially prestigious thing in any way: although I did brush up my CV and took some time composing a suitably keen-sounding cover letter, I had a reply to my application within a few hours, so I am assuming that there are not any very stringent entry requirements!  This is not the point though – the point is that it is the first mathematical event I am actually excited about.  Not just “looking forward to”…genuinely excited.  If I needed one more thing to dispel any remaining doubts about whether I am really suited to a career as a mathematician, then it is this –  the fact that I am thrilled about the prospect of attending what is essentially a series of lectures and exercise classes.

The reason I am looking forward to it so much is that the courses sound fascinating, they are closely related to my own speciality, and they are being taught by very well-respected mathematicians, all of whom I have heard of, most of whose work I have read, and some of whom I have even cited in my papers.  Having not done an undergraduate degree in mathematics, in every course I have done until now I have always felt as though I was frantically catching up, and in a state pf perpetual confusion, missing as I was large chunks of prerequisite knowledge.  For pretty much the first time, I will be able to study subjects that I am interested in, and that have direct relevance to what I am doing.  I will actually be able to follow the lectures!

So what is “Algebraic and Enumerative Combinatorics?”.  Well, clearly they are subfields of Combinatorics, and unfortunately Combinatorics is notoriously tricky to define.  Wikipedia does as best a job as could be expected in one line, by saying that it is the “study of finite or countable discrete structures”.  Countable here means in a one-to-one correspondence with the counting numbers 1,2,3,…, that is, the “smallest possible infinity”; and discrete means “separate and distinct” (it should not be confused with the alternate spelling!).

The “non-discrete” fields of mathematics are continuous subjects, which study objects which cannot be broken down into separate parts.  Another way of saying this is that, if we were to try to break down such an object, there would be uncountably many parts, in that between any two of them, there is always another one.  For example, the number of numbers between any two given numbers is uncountable, as is the number of points on a line (or the number of points between any two points on the line!).  The quintessential subjects of continuous mathematics are Analysis, Geometry, and Topology.  It may be handy to think about the difference between a digital clock (discrete) and an analogue clock (continuous, assuming we allow the second hand to take any value between seconds).

So, discreteness is certainly a defining feature of Combinatorics.  Finiteness/countability perhaps less so…for example there is no good reason not to study a graph having uncountably many vertices.  And really, no matter how you try to define Combinatorics, someone will probably be able to come up with an object which doesn’t neatly fit into it.  For example, if we state that Combinatorics is simply the study of discrete structures, then does Algebra count as Combinatorics?  Well, yes and no.  So no.  Take Group Theory for example – any finite group, such as the group of symmetry-preserving rotations of a polygon, is certainly a discrete object.  But then what about the group of rotations of a circle?  Well, then we are into continuous mathematics.  So in a sense, Algebra straddles the divide, meaning it is not a divide at all!  It is probably impossible to neatly classify mathematical subjects in this way, and I am no meta-mathematician (are there such people?) so I’ll leave it at that before I befuddle myself further.

As I said, Combinatorics is difficult to define.  There are however, certain defining characteristics…for example, Combinatorics tends to be wide rather than deep; its problems are usually easy to state (but not necessarily easy to solve!); its practitioners tend to be “problem-solvers” rather than “theory-builders”.  Often – and traditionally –  a combinatorial problem is one that involves counting something.  Indeed, this is exactly what Enumerative Combinatorics is: the counting of combinatorial objects.  Amusingly, when I did my MSc I was taught a course called Enumerative and Asymptotic Combinatorics by my current supervisor…he later told me he had originally wanted to call the course “Counting” but the university did not think this sounded sufficiently advanced! (Understandable, perhaps).   His notes for the course are still called, simply, “Notes on Counting”…google them if you’re interested!  (And if you’re still interested after that, his book on Combinatorics is highly recommended).

Increasingly, however, techniques from branches of continuous mathematics such as probability and analysis are used to investigate combinatorial structures. And as these applications become more entrenched, whole new branches of combinatorics develop: Probabilistic Combinatorics (as its name suggests, this is the application of probabilistic methods to combinatorial problems), Arithmetic Combinatorics (an especially difficult-to-define mix of number theory, harmonic analysis and combinatorics – this and its subfield Additive Combinatorics have been very fashionable ever since Ben Green and Terry Tao’s proof that there are arbitrarily long arithmetic progressions of prime numbers),  and of course, Algebraic Combinatorics.   Which finally gets me to my point!

Algebraic Combinatorics is, basically, what I do.  It involves the application of the vast body of algebraic theory to investigating problems about combinatorial objects (not to be confused with Combinatorial Algebra, which does the reverse!  As a general rule, if a mathematical subject consists of two words, then most of the time it involves using the first one to investigate the second).  To be precise, my research is in a subfield of Algebraic Combinatorics, known as Algebraic Graph Theory…as you might expect, the objects under consideration here are graphs (mathematicians are usually quite good at naming things sensibly).  As an example of what this is all about, take some graph, consisting of a number of dots joined by lines.    There are many algebraic structures which arise naturally from such an object; as many as there are ways in which to encode the structure of the graph algebraically.  We can express it as a matrix, and then bring the machinery of Linear Algebra to bear on it.  Or we could  encode some or all of its properties as a polynomial or other function in one or more variables.

My PhD research has been on algebraic properties of graph polynomials.  This sometimes involves studying structures which arise from structures which arise from structures.  For example, I have done some work on Galois groups of graph polynomials…that is, groups which arise naturally from polynomials which arise naturally from graphs.  This can go on and on!  And we can go back and forwards between subjects.  Combinatorics and Algebra are particularly rich in interactions of this kind.  For example we could take a graph, take the chromatic polynomial of that graph, compute the Galois group of that polynomial, form the power graph of that Galois group, etc!

The reason I like this kind of thing so much is that I like deep structure and theory in mathematics, of which there is an abundance in Algebra, but I am not particularly good at developing that theory (actually I haven’t even tried, but I wouldn’t know how to go about starting).  I am a problem-solver, but the way in which I like to solve problems is to abstract them, and to bring to bear on them the large bodies of theory that I love in mathematics (and that I have spent so long struggling to understand!).   Often this approach ends up becoming quite detached from the original subject (for example, it is not very clear what implications the Galois theory of graph polynomials have for the original graphs themselves), but as a pure mathematician, we needn’t concern ourselves with this kind of thing.  We can just do it for the sake of doing it, a bit like that old chestnut about why to climb Everest (“because it’s there”).  Perhaps I will have to start considering how I might justify this sort of thing however, if I am to subject myself to the whole “Impact”-based research evaluation thing in the future (see previous post-but-one).

Anyway, that’s quite enough for one day.  I meant to actually write about what we exactly we will be studying at this summer school!  But as usual I haven’t got past the basic definitions…oh well, perhaps next time.  And reading this back – apologies for all the parentheses.

]]> <![CDATA[Milliman Lecture III: Sum-product estimates, expanders, and exponential sums]]> http://terrytao.wordpress.com/2007/12/06/milliman-lecture-iii-sum-product-estimates-expanders-and-exponential-sums/ Fri, 07 Dec 2007 06:11:11 +0000 Terence Tao http://terrytao.wordpress.com/2007/12/06/milliman-lecture-iii-sum-product-estimates-expanders-and-exponential-sums/ This is my final Milliman lecture, in which I talk about the sum-product phenomenon in arithmetic combinatorics, and some selected recent applications of this phenomenon to uniform distribution of exponentials, expander graphs, randomness extractors, and detecting (sieving) almost primes in group orbits, particularly as developed by Bourgain and his co-authors.
In the previous two lectures we had concentrated on additive combinatorics – the study of additive operations and patterns on sets. Now we will look at arithmetic combinatorics – the simultaneous study of additive and multiplicative operations on sets; this is sort of a combinatorial analogue of commutative algebra.

There are many questions to study here, but the most basic is the sum-product problem, which we can state as follows. Let A be a finite non-empty set of elements of a ring R (e.g. finite sets of integers, or elements of a cyclic group {\Bbb Z}/q{\Bbb Z}, or sets of matrices over some ring). Then we can form the sum set

A+A := \{ a + b: a, b \in A \}

and the product set

A \cdot A := \{ a \cdot b: a, b \in A \}

To avoid degeneracies, let us assume that none (or very few) of the elements in A are zero divisors (as this may cause A \cdot A to become very small). Then it is easy to see that A+A and A \cdot A will be at least as large as A itself.

Typically, both of these sets will be much larger than A itself, indeed, if we select A at random, we generically expect A+A and A \cdot A to have cardinality comparable to |A|^2. But when A enjoys additive or multiplicative structure, the sets A+A or A \cdot A can be of size comparable to A. For instance, if A is an arithmetic progression \{a, a+r, a+2r, \ldots, a+(k-1)r\} or an additive subgroup in the ring R (modulo zero divisors, such as 0), then |A+A| \sim |A|. Similarly, if A is a geometric progression \{a, ar, ar^2, \ldots, ar^{k-1} \} or a multiplicative subgroup in the ring R, then |A \cdot A| \sim |A|. And of course, if A is both an additive and a multiplicative subgroup of R (modulo zero divisors), i.e. if A is a subring of R, then |A+A| and |A \cdot A| are both comparable in size to |A|. These examples are robust with respect to small perturbations; for instance, if A is a dense subset of an arithmetic progression or additive subgroup, then it is still the case that A+A is comparable in size to A. There are also slightly more complicated examples of interest, such as generalised arithmetic progressions, but we will not discuss these here.

Now let us work in the ring of integers {\Bbb Z}. This ring has no non-trivial finite additive subgroups or multiplicative subgroups (and it certainly has no non-trivial finite subrings), but it of course has plenty of arithmetic progressions and geometric progressions. But observe that it is rather difficult for a finite set A of integers to resemble both an arithmetic progression and a geometric progression simultaneously (unless A is very small). So one expects at least one of A+A and A \cdot A to be significantly larger than A itself. This claim was made precise by Erdős and Szemerédi, who showed that

\max( |A+A|, |A \cdot A| ) \gg |A|^{1+\varepsilon} (1)

for some absolute constant \varepsilon > 0. The value of this constant as improved steadily over the years; the best result currently is due to Solymosi, who showed that one can take \varepsilon arbitrarily close to 3/11. Erdős and Szemerédi in fact conjectured that one can take \varepsilon arbitrarily close to 1 (i.e. for any finite set of integers A, either the sum set or product set has to be very close to its maximal size of |A|^2), but this conjecture seems out of reach at present. Nevertheless, even just the epsilon improvement over the trivial bound of |A| is actually quite useful. It is the first example of what is now called the sum-product phenomenon: if a finite set A is not close to an actual subring, then either the sum set A+A or the product set A \cdot A must be significantly larger than A itself. One can view (1) as a “robust” version of the assertion that the integers contain no non-trivial finite subrings; (1) is asserting that in fact the integers contain no non-trivial finite sets which even come close to behaving like a subring.

In 1999, Tom Wolff posed the question of whether the sum-product phenomenon held true in finite fields {\Bbb F}_p of prime order (note that such fields have no non-trivial subrings), and in particular whether (1) was true when A \subset {\Bbb F}_p, and A was not close to being all of {\Bbb F}_p, in the sense that |A| \leq p^{1-\delta} for some \delta > 0; of course one would need \varepsilon to depend on \delta. (Actually, Tom only posed the question for |A| \sim p^{1/2}, being motivated by finite field analogues of the Kakeya problem, but the question was clearly of interest for other ranges of A as well.) This question was solved in the affirmative by Bourgain, Katz, and myself (in the range p^\delta \leq |A| \leq p^{1-\delta}) and then by Bourgain, Glibichuk, and Konyagin (in the full range 1 \leq |A| \leq p^{1-\delta}); the result is now known as the sum-product theorem for {\Bbb F}_p (and there have since been several further proofs and refinements of this theorem). The fact that the field has prime order is key; if for instance we were working in a field of order p^2, then by taking A to be the subfield of order p we see that both A+A and A \cdot A have exactly the same size as A. So any proof of the sum-product theorem must use at some point the fact that the field has prime order.

As in the integers, one can view the sum-product theorem as a robust assertion of the obvious statement that the field {\Bbb F}_p contains no non-trivial subrings. So the main difficulty in the proof is to find a proof of this latter fact which is robust enough to generalise to this combinatorial setting. The standard way to classify subrings is to use Lagrange’s theorem that the order of a subgroup divides the order of the whole group, which is proven by partitioning the whole group into cosets of the subgroup, but this argument is very unstable and does not extend to the combinatorial setting. But there are other ways to proceed. The argument of Bourgain, Katz, and myself (which is based on an earlier argument of Edgar and Miller), roughly speaking, proceeds by investigating the “dimension” of {\Bbb F}_p relative to A, or in other words the least number of elements v_1, \ldots, v_d in {\Bbb F}_p such that every element of {\Bbb F} can be expressed in the form a_1 v_1 + \ldots + a_d v_d. Note that the number of such representations is equal to |A|^d. The key observation is that as |{\Bbb F}_p| is prime, it cannot equal |A|^d if d > 1, and so by the pigeonhole principle some element must have more than one representation. One can use this “linear dependence” to reduce the dimension by 1 (assuming that A behaves a lot like a subring), and so can eventually reduce to the d=1 case, which is prohibited by our assumption A < p^{1-\delta}. (The hypothesis |A| > p^{\delta} is needed to ensure that the initial dimension d is bounded, so that the iteration only requires a bounded number of steps.) The argument of Bourgain, Glibichuk, and Konyagin uses a more algebraic method (a variant of the polynomial method of Stepanov), using the basic observation that the number of zeroes of a polynomial (counting multiplicity) is bounded by the degree of that polynomial to obtain upper bounds for various sets (such as the number of parallelograms in A). More recently, a short argument of Garaev proceeds using the simple observation that if A is any non-trivial subset of {\Bbb F}_p, then there must exist a \in A such that a+1 \not \in A; applying this to the “fraction field” Q[A] := \{ (a-b)/(c-d): a,b,c,d \in A, c \neq d \} of A one can conclude that Q[A] does not in fact behave like a field, and hence A does not behave like a ring.

The sum-product phenomenon implies that if a set A \subset {\Bbb F}_p of medium size p^\delta \leq |A| \leq p^{1-\delta} is multiplicatively structured (e.g. it is a geometric progression or a multiplicative subgroup) then it cannot be additively structured: A+A is significantly larger than A. It turns out that with a little bit of extra work, this observation can be iterated: A+A+A+A is even larger than A, and so on and so forth, and in fact one can show that kA = {\Bbb F}_p for some bounded k depending only on \delta, where kA := A + \ldots + A is the k-fold sumset of A. (The key to this iteration essentially lies in the inclusion (kA) \cdot (kA) \subset k^2 (A^2), which is a consequence of the distributive law. The use of this law unfortunately breaks the symmetry between multiplication and addition that one sees in the sum-product estimates.) Thus any multiplicatively structured subset A of {\Bbb F}_p of medium size must eventually additively generate the whole field. As a consequence of this, one can show that A is an additive expander, which roughly speaking means that A+B is spread out on a significantly larger set than B for any medium-sized set B. (In more probabilistic language, if one considered the random walk whose steps were drawn randomly from A, then this walk would converge extremely rapidly to the uniform distribution.) From that observation (and some more combinatorial effort), one can in fact conclude that multiplicatively structured sets must be distributed uniformly in an additive sense; if they concentrated too much in, say, a subinterval of {\Bbb F}_p, then this could be used to contradict the additive expansion property.

Let me note one cute application of this technology, due to Bourgain, to the Diffie-Hellman key exchange protocol and its relatives in cryptography. Suppose we have two people, Alice and Bob, who want to communicate privately and securely, but have never met each other before and can only contact each other via an unsecured network (e.g. the internet, or physical mail), in which anyone can eavesdrop. How can Alice and Bob achieve this?

If one was sending a physical object (e.g. a physical letter) by physical mail (which could be opened by third parties), one could proceed as follows.

  1. Alice places the object in a box, and locks the box with her own padlock, keeping the key. She then mails the locked box to Bob. Anyone who intercepts the box cannot open it, since they don’t have Alice’s key.
  2. Of course, Bob can’t open the box either. But what he can do instead is put his own padlock on the box (keeping the key), and sends the doubly locked box back to Alice.
  3. Alice can’t unlock Bob’s padlock… but she can unlock her own. So she removes her lock, and sends the singly locked box back to Bob.
  4. Bob can unlock his own padlock, and so retreives the object safely. At no point was the object available to any interceptor.

A similar procedure (a slight variant of the Diffie-Hellman protocol, essentially the Massey-Omura cryptosystem) can be used to transmit a digital message g (which one should think of as just being a number) from Alice to Bob over an unsecured network, as follows:

  1. Alice and Bob agree (over the unsecured network) on some large prime p (larger than the maximum size of the message g).
  2. Alice “locks” the message g by raising it to a power a mod p, where Alice generates the “key” a randomly and keeps it secret. She then sends the locked message g^a \hbox{ mod } p to Bob.
  3. Bob can’t decode this message (he doesn’t know a), but he doubly locks the message by raising the message to his own power b, and returns the doubly locked message g^{ab} \hbox{ mod } p back to Alice.
  4. Alice then “unlocks” her part of the message by taking the a^{th} root (which can be done by Cauchy’s theorem) and sends g^b \hbox{ mod } p back to Bob.
  5. Bob then takes the b^{th} root of the message and recovers g.

An eavesdropper (let’s call her Eve) could intercept p, as well as the three “locked” values g^a, g^b, g^{ab} \hbox{ mod } p, but does not directly recover g. Now, it is possible that one could use this information to reconstruct g (indeed, if one could quickly take discrete logarithms, then this would be a fairly easy task) but no feasible algorithm for this is known (if p is large, e.g. 500+ digits); the problem is generally believed to be roughly comparable in difficulty to that of factoring large numbers. But no-one knows how to rigorously prove that the Diffie-Hellman reconstruction problem is hard (e.g. non-polynomial time); indeed, this would imply P \neq NP, since this reconstruction problem is easily seen to be in NP (though it is not believed to be NP-complete).

Using the sum-product technology, Bourgain was at least able to show that the Diffie-Hellman protocol was secure (for sufficiently large p) if Eve was only able to see the high bits of g^a, g^b, g^{ab} \hbox{ mod } p, thus pinning down g^a, g^b, g^{ab} to intervals. The reason for this is that the set \{ (g^a, g^b, g^{ab}) \in {\Bbb F}_p^3: a,b \in {\Bbb Z} \} has a lot of multiplicative structure (indeed, it is a multiplicative subgroup of the ring {\Bbb F}_p^3) and so should be uniformly distributed in an additive sense (by adapting the above sum-product technology to {\Bbb F}_p^3).

Another application of sum-product technology was to build efficient randomness extractors – deterministic algorithms that can create high-quality (very uniform) random bits from several independent low-quality (non-uniform) random sources; such extractors are of importance in computer science and cryptography. Basically, the sum-product estimate implies that if A, B, C \subset {\Bbb F}_p are sets of medium size, then the set A+B \cdot C is significantly larger than A, B, or C. As a consequence, if X, Y, Z are independent random variables in {\Bbb F}_p which are not too narrowly distributed(in particular, they are not deterministic, and thus distributed only on a single value), one can show (with the assistance of some additive combinatorics) that the random variable X+YZ is significantly more uniformly distributed than X, Y, or Z. Iterating this leads to some surprisingly good randomness extractors, as was first observed by Barak, Impagliazzo, and Wigderson.

Another application of the above sum-product technology was to get a product estimate in matrix groups, such as SL_2({\Bbb F}_p). Indeed, Helfgott was able to show that if A was a subset of SL_2({\Bbb F}_p) of medium or small size, and it was not trapped inside a proper subgroup of SL_2({\Bbb F}_p), then A \cdot A \cdot A was significantly larger than A itself. (One needs to work with triple products here instead of double products for a rather trivial reason: if A was the union of a subgroup and some external element, then A \cdot A is still comparable in size to A, but A \cdot A \cdot A will be much larger. This result may not immediately look like a sum-product estimate, because there is no obvious addition, but it is concealed within the matrix multiplication law for SL_2({\Bbb F}_p). The key observation in Helfgott’s argument, which relies crucially on the sum-product estimate, is that if V is a collection of diagonal matrices in SL_2({\Bbb F}_p) of medium size, and g is a non-diagonal matrix element, then the set \hbox{tr}(V g V g^{-1}) is significantly larger than V itself. If one works out explicitly what this trace is, one sees a sum-product type of result emerging. Conversely, if the trace \hbox{tr}(A) of a group-like set A is large, then the conjugacy classes in A are fairly small (since trace is conjugation-invariant), which forces many pairs in A to commute, which creates large sets V of simultaneously commuting (and hence simultaneously diagonalisable) elements, due to the fact (specific to SL_2) that if two elements commute with a third, then they are quite likely to commute with each other. The tension between these two implications is what underlies Helfgott’s results.

The estimate of Helfgott shows that multiplication by medium-size sets in SL_2({\Bbb F}_p) expands rapidly across the group (unless it is trapped in a subgroup). As a consequence of Helfgott’s estimate, Bourgain and Gamburd were able to show that if S was any finite symmetric set of matrices in SL_2({\Bbb Z}) which generated a sufficiently large (or more precisely, Zariski dense) subgroup of SL_2({\Bbb Z}), and S_p was the projection of S to SL_2({\Bbb Z}_p), then the random walk using S_p on SL_2({\Bbb Z}_p) was very rapidly mixing, so that after about O(\log p) steps, the walk was very close to uniform. (The precise statement was that the Cayley graph associated to S_p for each p formed an expander family.) Quite recently, Bourgain, Gamburd, and Sarnak have applied these results (and generalisations thereof) to the problem of detecting (or sieving) almost primes in thin algebraically generated sets. To motivate the problem, we observe that many classical questions in prime number theory can be rephrased as one of detecting prime points (p_1,\ldots,p_d) \in {\mathcal P}^d in algebraic subsets {\mathcal O} of a lattice {\Bbb Z}^d. For instance, the twin prime problem asks whether the line {\mathcal O} = \{ (n,n+2) \in {\Bbb Z}^2 \} contains infinitely many prime points. In general, these problems are very difficult, especially once one considers sets described by polynomials rather than linear functions; even the one-dimensional problem of determining whether the set {\mathcal O} = \{ n^2+1: n \in {\Bbb Z} \} contains infinitely many primes has been open for quite a long time (though it is worth mentioning the celebrated result of Friedlander and Iwaniec that the somewhat larger set {\mathcal O} = \{ n^2 + m^4: n,m \in {\Bbb Z} \} is known to have infinitely many primes).

So prime points are hard to detect. However, by using methods from sieve theory, one can often detect almost prime points in various sets {\mathcal O} – points whose coordinates are the products of only a few primes. For instance, a famous theorem of Chen shows that the line {\mathcal O} = \{ (n,n+2) \in {\Bbb Z}^2 \} contains infinitely many points which are almost prime in the sense that the first coordinate is prime, and the second coordinate is the product of at most two primes. The basic idea of sieve theory is to sift out primes and almost primes by removing all points whose coordinates are divisible by small factors (and then, due to various generalisations of the inclusion-exclusion principle, one has to add back in points which are divisible by multiple small factors, and so forth). See my post on sieve theory and the parity problem for further discussion. In order for sieve theory to work well, one needs to be able to accurately count the size of the original set {\mathcal O} (or more precisely, the size of this set restricted to a ball or a similar object), and also need to count how many points in that set have a certain residue class modulo q, for various values of q. (For instance, to sieve out twin primes or twin almost primes in the interval {1,..,N}, one needs to count how many elements n in that interval are such that n and n+2 are both invertible modulo q (i.e. coprime to q) for various values of q.)

For arbitrary algebraic sets {\mathcal O}, these tasks are very difficult. For instance, even the basic task of determining whether a set {\mathcal O} described by several polynomials is non-empty is essentially Hilbert’s tenth problem, which is undecidable in general. But if the set {\mathcal O} is generated by a group \Lambda acting on {\Bbb Z}^d (in some polynomial fashion), thus {\mathcal O} = \Lambda b for some point b \in {\Bbb Z}^d, then the problems become much more tractable. If the group \Lambda is generated by some finite set S, and we restrict attention to group elements with some given word length, the problem of understanding how {\mathcal O} is distributed modulo q is equivalent to asking how random walks on S of a given length distribute themselves on ({\Bbb Z}/q{\Bbb Z})^d. This latter problem is very close to the problem solved by the mixing results of Bourgain and Gamburd mentioned earlier, which is where the link to sum-product estimates arises from. Indeed, Bourgain, Gamburd, and Sarnak have now shown that rather general classes of algebraic sets generated by subgroups of SL_2({\Bbb Z}) will contain infinitely many almost primes, as long as there are no obvious algebraic obstructions; the methods should hopefully extend to more general groups, such as subgroups of SL_n({\Bbb Z}).

[Update, Dec 7: terminology and typos corrected.]

]]>