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 , or sets of matrices over some ring). Then we can form the sum set

and the product set

To avoid degeneracies, let us assume that none (or very few) of the elements in A are zero divisors (as this may cause to become very small). Then it is easy to see that and 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 and to have cardinality comparable to . But when A enjoys additive or multiplicative structure, the sets or can be of size comparable to A. For instance, if A is an arithmetic progression or an additive subgroup in the ring R (modulo zero divisors, such as 0), then . Similarly, if A is a geometric progression or a multiplicative subgroup in the ring R, then . 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 and 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 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 . 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 and to be significantly larger than A itself. This claim was made precise by Erdős and Szemerédi, who showed that

(1)

for some absolute constant . 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 arbitrarily close to 3/11. Erdős and Szemerédi in fact conjectured that one can take 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 ), 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 or the product set 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 of prime order (note that such fields have no non-trivial subrings), and in particular whether (1) was true when , and A was not close to being all of , in the sense that for some ; of course one would need to depend on . (Actually, Tom only posed the question for , 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 ) and then by Bourgain, Glibichuk, and Konyagin (in the full range ); the result is now known as the sum-product theorem for (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 , then by taking A to be the subfield of order p we see that both and 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 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 relative to A, or in other words the least number of elements in such that every element of can be expressed in the form . Note that the number of such representations is equal to . The key observation is that as is prime, it cannot equal if , 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 . (The hypothesis 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 , then there must exist such that ; applying this to the “fraction field” 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 of medium size is multiplicatively structured (e.g. it is a geometric progression or a multiplicative subgroup) then it cannot be additively structured: 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 for some bounded k depending only on , where is the k-fold sumset of A. (The key to this iteration essentially lies in the inclusion , 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 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 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 , 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 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 back to Alice.
4. Alice then “unlocks” her part of the message by taking the root (which can be done by Cauchy’s theorem) and sends back to Bob.
5. Bob then takes the root of the message and recovers g.

An eavesdropper (let’s call her Eve) could intercept p, as well as the three “locked” values , 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 , 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 , thus pinning down to intervals. The reason for this is that the set has a lot of multiplicative structure (indeed, it is a multiplicative subgroup of the ring ) and so should be uniformly distributed in an additive sense (by adapting the above sum-product technology to ).

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 are sets of medium size, then the set is significantly larger than A, B, or C. As a consequence, if X, Y, Z are independent random variables in 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 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 . Indeed, Helfgott was able to show that if A was a subset of of medium or small size, and it was not trapped inside a proper subgroup of , then was significantly larger than A itself. (One needs to work with triple products here instead of double products for a rather trivial reason: if was the union of a subgroup and some external element, then is still comparable in size to A, but 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 . 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 of medium size, and g is a non-diagonal matrix element, then the set 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 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 ) 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 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 which generated a sufficiently large (or more precisely, Zariski dense) subgroup of , and was the projection of S to , then the random walk using on was very rapidly mixing, so that after about steps, the walk was very close to uniform. (The precise statement was that the Cayley graph associated to 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 in algebraic subsets of a lattice . For instance, the twin prime problem asks whether the line 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 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 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 – points whose coordinates are the products of only a few primes. For instance, a famous theorem of Chen shows that the line 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 (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 , these tasks are very difficult. For instance, even the basic task of determining whether a set described by several polynomials is non-empty is essentially Hilbert’s tenth problem, which is undecidable in general. But if the set is generated by a group acting on (in some polynomial fashion), thus for some point , then the problems become much more tractable. If the group is generated by some finite set S, and we restrict attention to group elements with some given word length, the problem of understanding how is distributed modulo q is equivalent to asking how random walks on S of a given length distribute themselves on . 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 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 .

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

In this post, when I talk about a graph, I mean an abstract collection of vertices V, together with some abstract edges E joining pairs of vertices together. We will assume that the graph is undirected (the edges do not have a preferred orientation), loop-free (an edge cannot begin and start at the same vertex), and multiplicity-free (any pair of vertices is joined by at most one edge). More formally, we can model all this by viewing E as a subset of , the set of 2-element subsets of V, and we view the graph G as an ordered pair G = (V,E). (The notation is set up so that .)

Now one of the great features of graphs, as opposed to some other abstract maths concepts, is that they are easy to draw: the abstract vertices become dots on a plane, while the edges become line segments or curves connecting these dots. [To avoid some technicalities we do not allow these curves to pass through the dots, except if the curve is terminating at that dot.] Let us informally refer to such a concrete representation D of a graph G as a drawing of that graph. Clearly, any non-trivial graph is going to have an infinite number of possible drawings. In some of these drawings, a pair of edges might cross each other; in other drawings, all edges might be disjoint (except of course at the vertices, where edges with a common endpoint are obliged to meet). If G has a drawing D of the latter type, we say that the graph G is planar.

Given an abstract graph G, or a drawing thereof, it is not always obvious as to whether that graph is planar; just because the drawing that you currently possess of G contains crossings, does not necessarily mean that all drawings of G do. The wonderful little web game “Planarity” illustrates this point excellently. Nevertheless, there are definitely graphs which are not planar; in particular the complete graph on five vertices, and the complete bipartite graph on two sets of three vertices, are non-planar.

There is in fact a famous theorem of Kuratowski that says that these two graphs are the only “source” of non-planarity, in the sense that any non-planar graph contains (a subdivision of) one of these graphs as a subgraph. (There is of course the even more famous four-colour theorem that asserts that every planar graphs is four-colourable, but this is not the topic of my post today.)

Intuitively, if we fix the number of vertices |V|, and increase the number of edges |E|, then the graph should become “increasingly non-planar”; conversely, if we keep the same number of edges |E| but spread them amongst a greater number of vertices |V|, then the graph should become “increasingly planar”. Is there a quantitative way to measure the “non-planarity” of a graph, and to formalise the above intuition as some sort of inequality?

It turns out that there is an elegant inequality that does precisely this, known as the crossing number inequality. It was first discovered by Ajtai-Chvátal-Newborn-Szemerédi and by Leighton (the latter being motivated by optimising VLSI designs). Nowadays it can be proven by two elegant amplifications of Euler’s formula, as we shall see.

If D is a drawing of a graph G, we define cr(D) to be the total number of crossings – where pairs of edges intersect at a point, for a reason other than sharing a common vertex. (If multiple edges intersect at the same point, each pair of edges counts once.) We then define the crossing number cr(G) of G to be the minimal value of cr(D) as D ranges over the drawings of G. Thus for instance cr(G)=0 if and only if G is planar. One can also verify that the two graphs and have a crossing number of 1. This quantity cr(G) will be the measure of how non-planar our graph G is. The problem is to relate this quantity in terms of the number of vertices |V| and the number of edges |E|. We of course do not expect an exact identity relating these three quantities (two graphs with the same number of vertices and edges may have a different number of crossing numbers), so will settle for good upper and lower bounds on cr(G) in terms of |V| and |E|.

How big can the crossing number of a graph G = (V,E) be? A trivial upper bound is cr(G) = O( |E|^2 ), because if we place the vertices in general position (or on a circle) and draw the edges as line segments, then every pair of edges crosses at most once. But this bound does not seem very tight; we expect to be able to find drawings in which most pairs of edges in fact do not intersect.

Let’s turn our attention instead to lower bounds. We of course have the trivial lower bound ; can we do better? Let’s first be extremely unambitious and see when one can get the minimal possible improvement on this bound, namely . In other words, we want to find some conditions on |V| and |E| which will force G to be non-planar. We can turn this around by taking contrapositives: if G is planar, what does this tell us about |V| and |E|?

Here, the natural tool is Euler’s formula |V|-|E|+|F|=2, valid for any planar drawing, where |F| is the number of faces (including the unbounded face). [This is the one place where we shall really use the topological structure of the plane; the rest of the argument is combinatorial.  There are some minor issues if the graph is disconnected, or if there are vertices of degree one or zero, but these are easily dealt with.] What do we know about |F|? Well, every face is adjacent to at least three edges, whereas every edge is adjacent to exactly two faces. By double counting the edge-face incidences, we conclude that . Eliminating |F|, we conclude that for all connected planar graphs with at least one cycle (and this bound is tight when the graph is triangular), which then implies in general. Taking contrapositives, we conclude

whenever . (*)

Now, let us amplify this inequality by exploiting the freedom to delete edges. Indeed, observe that if a graph G can be drawn with only cr(G) crossings, then we can delete one of the crossings by removing an edge associated to that crossing, and so we can remove all the crossings by deleting at most cr(G) edges, leaving at least |E|-cr(G) edges (and |V| vertices). Combining this with (*) we see that regardless of the number of crossings, we have

leading to the following amplification of (*):

. (**)

This is not the best bound, though, as one can already suspect by comparing (**) with the crude upper bound . We can amplify (**) further by exploiting a second freedom, namely the ability to delete vertices. One could try the same sort of trick as before, deleting vertices which are associated to a crossing, but this turns out to be very inefficient (because deleting vertices also deletes an unknown number of edges, many of which had nothing to do with the crossing). Indeed, it would seem that one would have to be fiendishly clever to find an efficient way to delete a lot of crossings by deleting only very few vertices.

However, there is an amazing (and unintuitive) principle in combinatorics which states that when there is no obvious “best” choice for some combinatorial object (such as a set of vertices to delete), then often trying a random choice will give a reasonable answer, if the notion of “random” is chosen carefully. (See this paper of Gowers for some further discussion of this principle.) The application of this principle is known as the probabilistic method, first introduced by Erdős.

Here is how it works in this current setting. Let be a parameter to be chosen later. We will randomly delete all but a fraction p of the vertices, by letting each vertex be deleted with an independent probability of 1-p (and thus surviving with a probability of p). Let V’ be the set of vertices that remain. Once one deletes vertices, one also has to delete the edges attached to these vertices; let E’ denote the surviving edges (i.e. the edges connected to vertices in V’). Let G’=(V’,E’) be the surviving graph (known as the subgraph of G induced by V’). Then from (**) we have

.

Now, how do we get from this back to the original graph G = (V,E)? The quantities |V’|, |E’|, and cr(G’) all fluctuate randomly, and are difficult to compute. However, their expectations are much easier to compute. Accordingly, we take expectations of both sides (this is an example of the first moment method). Using linearity of expectation, we have

These quantities are all relatively easy to compute. The easiest is . Each vertex in V has a probability p of ending up in V’, and thus contributing 1 to |V’|. Summing up (using linearity of expectation again), we obtain .

The quantity is almost as easy to compute. Each edge e in E will have a probability of ending up in E’, since both vertices have an independent probability of p of surviving. Summing up, we obtain . (The events that each edge ends up in E’ are not quite independent, but the great thing about linearity of expectation is that it works even without assuming any independence.)

Finally, we turn to . Let us draw G in the optimal way, with exactly cr(G) crossings. Observe that each crossing involves two edges and four vertices. (If the two edges involved in a crossing share a common vertex as well, thus forming an shape, one can reduce the number of crossings by 1 by swapping the two halves of the loop in the shape. So with the optimal drawing, the edges in a crossing do not share any vertices in common.) Passing to G’, we see that the probability that the crossing survives in this drawing is only . By one last application of linearity of expectation, the expected number of crossings of this diagram that survive for G’ is . This particular diagram may not be the optimal one for G’, so we end up with an inequality . Fortunately for us, this inequality goes in the right direction, and we get a useful inequality:

.

In terms of cr(G), we have

.

To finish the amplification, we need to optimise in p, subject of course to the restriction , since p is a probability. One can solve the optimisation problem exactly, but we will perform a cheaper computation by settling for a bound which is close to the optimal bound rather than exactly equal to it. A general principle is that optima are often obtained when two of the terms are roughly in balance. A bit of thought reveals that it might be particularly good to have just barely smaller than . (If it is a lot smaller, then p will be large, and we don’t get a good bound on the right. If instead is a lot bigger, then we are likely to have a negative right-hand side.) For instance, we could choose p so that ; this is legal as long as . Substituting this we obtain the crossing number inequality

whenever . (***)

This is quite a strong amplification of (*) or (**) (except in the transition region in which |E| is comparable to |V|).

Is it sharp? We can compare it against the trivial bound , and we observe that the two bounds match up to constants when |E| is comparable to . (Clearly, |E| cannot be larger than .) So the crossing number inequality is sharp (up to constants) for dense graphs, such as the complete graph on n vertices.

Are there any other cases where it is sharp? We can answer this by appealing to the symmetries of (***). By the nature of its proof, the inequality is basically symmetric under passage to random induced subgraphs, but this symmetry does not give any further examples, because random induced subgraphs of dense graphs again tend to be dense graphs (cf. the computation of and above). But there is a second symmetry of (***) available, namely that of replication. If one takes k disjoint copies of a graph G = (V,E), one gets a new graph with k|V| vertices and k|E| edges, and a moment’s thought will reveal that the new graph has a crossing number of k cr(G). Thus replication is a symmetry of (***). Thus, (***) is also sharp up to constants for replicated dense graphs. It is not hard to see that these examples basically cover all possibilities of |V| and |E| for which . Thus the crossing number inequality cannot be improved except for the constants. (The best constants known currently can be found in a recent paper of Pach, Radoicic, Tardos, and Tóth.)

[A general principle, by the way, is that one can roughly gauge the "strength" of an inequality by the number of independent symmetries (or approximate symmetries) it has. If for instance there is a three-parameter family of symmetries, then any example that demonstrates that sharpness of that inequality is immediately amplified to a three-parameter family of such examples (unless of course the example is fixed by a significant portion of these symmetries). The more examples that show an inequality is sharp, the more efficient it is - and the harder it is to prove, since one cannot afford to lose anything (other than perhaps some constants) in every one of the sharp example cases. This principle is of course consistent with my previous post on arbitraging a weak asymmetric inequality into a strong symmetric one.]

– Application: the Szemerédi-Trotter theorem –

It was noticed by Székely that the crossing number is powerful enough to give easy proofs of several difficult inequalities in combinatorial incidence geometry. For instance, the Szemerédi-Trotter theorem concerns the number of incidences between a finite collection of points P and lines L in the plane. For instance, the three lines and three points of a triangle form six incidences; the five lines and ten points of a pentagram form 20 incidences; and so forth.

One can ask the question: given |P| points and |L| lines, what is the maximum number of incidences I(P,L) one can form between these points and lines? (The minimum number is obviously 0, which is a boring answer.) The trivial bound is , but one can do better than this, because it is not possible for every point to lie on every line. Indeed, if we use nothing more than the axiom that every two points determine at most one line, combined with the Cauchy-Schwarz inequality, it is not hard to show (by double-counting the space of triples such that ) that

. (****)

Dually, using the axiom that two lines intersect in at most one point, we obtain

.

(One can also deduce one inequality from the other by projective duality.)

Can one do better? The answer is yes, if we observe that a configuration of points and lines naturally determines a drawing of a graph, to which the crossing number can be applied. To see this, assume temporarily that every line in L is incident to at least two points in P. A line l in L which is incident to k points in P will thus contain k-1 line segments in P; k-1 is comparable to k. Since the sum of all the k is I(P,L) by definition, we see that there are roughly I(P,L) line segments of L connecting adjacent points in P; this is a diagram with |P| vertices and roughly |I(P,L)| edges. On the other hand, a crossing in this diagram can only occur when two lines in L intersect. Since two lines intersect in at most one point, the total number of crossings is O(|L|^2). Applying the crossing number inequality (***), we obtain

if I(P,L) is much larger than |P|

which leads to

.

We can then remove our temporary assumption that lines in L are incident to at least two points, by observing that lines that are incident to at most one point will only contribute O(|L|) incidences, leading to the Szemerédi-Trotter theorem

.

This bound is somewhat stronger than the previous bounds, and is in fact surprisingly sharp; a typical example that demonstrates this is when P is the lattice and L is the set of lines with slope and intercept ; here and the number of incidences is roughly .

The original proof of this theorem, by the way, proceeded by amplifying (****) using the method of cell decomposition; it is thus somewhat similar in spirit to Szekély’s proof, but was a bit more complicated technically. Wolff conjectured a continuous version of this theorem for fractal sets, sometimes called the Furstenberg set conjecture, and related to the Kakeya conjecture; a small amount of progress beyond the analogue of (****) is known, thanks to work of Katz and myself and of Bourgain, but we are still far from the best possible result here.

– Application: sum-product estimates –

One striking application of the Szemerédi-Trotter theorem (and by extension, the crossing number inequality) is to the arena of sum-product estimates in additive combinatorics, which is currently a very active area of research, especially in finite fields, due to its connections with some long-standing problems in analytic number theory, as well as to some computer science problems concerning randomness extraction and expander graphs. However, our focus here will be on the more classical setting of sum-product estimates in the real line .

Let A be a finite non-empty set of non-zero real numbers. We can form the sum set

and the product set

.

If A is in “general position”, it is not hard to see that and both have cardinality comparable to . However, in certain cases one can make one or the other sets significantly smaller. For instance, if A is an arithmetic progression , then the sum set has cardinality comparable to just |A|. Similarly, if A is a geometric progression , then the product set has cardinality comparable to |A|. But clearly A cannot be an arithmetic progression and a geometric progression at the same time (unless it is very short). So one might conjecture that at least one of the sum set and product set should be significantly larger than A. Informally, this is saying that no finite set of reals can behave much like a subring of .

This intuition was made precise by Erdős and Szemerédi, who established the lower bound

for some small which they did not make explicit. They then conjectured that in fact c should be made arbitrary close to the optimal value of 1, and more precisely that

for large |A| and some absolute constant .

The Erdős-Szemerédi conjecture remains open, however the value of c has been improved; currently, the best bound is due to Solymosi, who showed that c can be arbitrarily close to 3/11. Solymosi’s argument is based on an earlier argument of Elekes, who obtained c=1/4 by a short and elegant argument based on the Szemerédi-Trotter theorem which we will now present. The basic connection between the two problems stems from the familiar formula y=mx+b for a line, which clearly encodes a multiplicative and additive structure. We already used this connection implicitly in the example that demonstrated that the Szemerédi-Trotter theorem was sharp. For Elekes’ argument, the challenge is to show that if and are both small, then a suitable family of lines associated to A will have a high number of incidences with some set of points associated to A, so that the Szemerédi-Trotter may then be profitably applied. It is not immediately obvious exactly how to do this, but Elekes settled upon the choice of letting , and letting L be the space of lines y=mx+b with slope in and intercept in A, thus and . One observes that each line in L is incident to |A| points in P, leading to incidences. Applying the Szemerédi-Trotter theorem and doing the algebra one eventually concludes that . (A more elementary proof of this inequality, not relying on the Szemerédi-Trotter theorem or crossing number bounds, and thus having the advantage on working on other archimedean fields such as , was subsequently found by Solymosi, but the best bounds on the sum-product problem in still rely very much on the Szemerédi-Trotter inequality.)