Dynamic numbers - the work of Elon Lindenstrauss

Elon Lindenstrauss got the Fields Medal for developing tools in the area of dynamical systems and using them to crack hard problems in number theory.

Elon Lindenstrauss, Princeton University
Fields medallist 2010.

As the name suggests, number theory studies the basic properties of numbers. The whole numbers 1, 2, 3, etc are probably the first thing that spring to mind when you think about numbers. Close to follow are the rational numbers: these are the fractions, numbers of the form $p/q$, where $p$ and $q$ are both whole numbers. But there also irrational numbers, which can't be written as fractions. An example is the number $\pi$: some people write it as 22/7, but that's just an approximation: it's close to $\pi$, but not exactly equal to it. In fact, there isn't any fraction that's exactly equal to $\pi$.

In turns out that you can approximate an irrational number, call it $\alpha$, by a fraction to any degree of accuracy. If you give me a really small number $\epsilon$, then no matter how small $\epsilon$ is, I can find you a fraction that's within $\epsilon$ of $\alpha$. But some approximations are better than others. The fraction 2147865/68341 is a tiny bit closer to $\pi$ than 22/7, but it's also much more horrible to write down because it has such a large denominator (and as a result a very large numerator). So what's the ideal relationship between the accuracy of approximation and the denominator of a fraction?

In the 19th century the German mathematician Johan Dirichilet came up with a notion of this ideal relationship. He decided that an approximation $p/q$ of an irrational number $\alpha$ should be no further from $\alpha$ than $1/q^2$. In other words, if the denominator $q$ is large, (so that $q^2$ is even larger and therefore $1/q^2$ very small), then the fraction should make up for this by being close enough (within $1/q^2$) of $\alpha$. Dirichilet proved, and the proof wasn't very hard, that given any irrational number $\alpha$, you can always find infinitely many fractions $p/q$ which satisfy this criterion. So there's a "nice" approximation, in Dirichilet's sense, for any level of accuracy.

It turns out that something similar is true for pairs of irrational numbers $\alpha$ and $\beta$. There are infinitely many fractions $p/q$ and $r/q$ which are nice simultaneous approximations of $\alpha$ and $\beta$: the difference between $\alpha$ and $p/q$ times the difference between $\beta$ and $r/q$ is less than $1/q^3.$ Put in the form of an equation, this is $$\vert \alpha-p/q \vert \times \vert \beta - r/q \vert \frac{1}{q^3}.$$ Since pairs of numbers can be interpreted as the coordinates of a point on a 2D plane, this result gives a measure of how well points with irrational coordinates can be approximated using points with rational coordinates that have the same denominator.

In the twentieth century the mathematician John Littlewood decided that we should be able to do even better than this. Given any two irrational numbers $\alpha$ and $\beta$ and an $\epsilon$ that's as small as you like, there should be fractions $p/q$ and $r/q$ so that $$\vert \alpha - p/q \vert \times \vert \beta - r/q \vert \frac{\epsilon}{q^3}.$$ The statement seemed like an easy generalisation, but no-one has so far been able to prove it. It's become known as the Littlewood conjecture.

Number theory is littered with statements that look like they should be easy to prove but turn out to be incredibly hard. In these cases you have to look for clever tools to help you find a solution. In his work Elon Lindenstrauss did just that, using tools from dynamical systems theory. As an example of a dynamical system, think of the 2D plane in which every point is defined by its co-ordinates, a pair of numbers $(x,y)$. Now take any such point $(x,y)$ and shift it by a certain distance $\alpha$ to the right and up by another distance $\beta$. This rule gives you a dynamical system. You can apply it again and again and see what happens to the trajectories of various points.

In the case of the plane, nothing very interesting happens, as trajectories just move further and further away from the centre of the plane, given by the coordinates $(0,0)$. If, however, if you look at the surface of a doughnut, things get more interesting. You can make such a surface by taking a square from the plane, turning it into a cylinder by gluing together the left and right edges, and then bending it around and gluing together the circles on either end of the cylinder. In this way, the doughnut's surface inherits the coordinates defined on the original square. Things now become more interesting as you shift points around as before, using the numbers $\alpha$ and $\beta$. Trajectories can travel round and round and visit the same patch of doughnut lots of times.

It turns out that if your two numbers $\alpha$ and $\beta$ are irrational, then the dynamical system is what's called ergodic: loosely speaking, trajectories will visit every patch of the doughnut surface and patches of equal area will see comparable rates of traffic. And here is the connection with the Littlewood conjecture: suppose that the pair of numbers $\alpha$ and $\beta$, the distances by which you're shifting points, are the pair of irrational numbers you're trying to simultaneous approximate by fractions. It turns out that proving the Littlewood conjecture is equivalent to showing that you can get every point $(x,y)$ sufficiently close to the point $(0,0)$, just by shifting along using the numbers $\alpha$ and $\beta$ a suitable number of times. The number of times you need to shift along gives you the denominator $q$ you're after.

Using a more complicated dynamical system, Lindenstrauss and his colleagues made massive progress towards a proof of the elusive Littlewood conjecture. They showed that if there are any pairs of numbers $(\alpha, \beta)$ that can't be approximated in the nice way stipulated by the conjecture, then they make up only a negligible portion of the plane in which they live. There are pairs for which the conjecture isn't yet proven, in fact there's infinitely many of them, but as Lindenstrauss showed, collectively they are nothing more than drops in the ocean of the 2D plane.

It's this progress on Littlewood's conjecture that forms part of the body of work for which Lindenstrauss is being honoured. You can find out more about his work in this excellent description on the ICM website.