In this post we describe a precise achievement of Lindenstrauss and his co-authors, to get a more concrete idea of what is his research about. There are several deep contributions which one could try to explain, but it seems that the simplest to formulate is the one related to Littlewood’s conjecture, so let’s focus on this one.
Diophantus of Alexandria lived in the 3rd century ad, approx. He is best known by his Arithmetica, a collection of 13 books focused on algebraic manipulations and problems about whole numbers. It is now a common practice to use the term Diophantine to refer to any technique which is mainly concerned about integers. For instance, a Diophantine equation is an equation to be solved in the integers (or the rationals, or even number fields depending on the context). Diophantine approximation refers to a part of number theory where the main interest is in studying approximations that involve integers or rational numbers. Here is the most basic, elementary, and often useful result that can be labeled as a Diophantine approximation theorem:
Proposition. Let n be an integer. If n is not zero, then |n| is at least 1.
Approximating irrationals by rational numbers.
The set of rational numbers is dense in the set of real numbers. That is, for every real number x we can find rational approximations as close to x as we want. However, some rational approximations are better than others even if they have more or less the same error. For instance, take x=√2=1.4142135… and look at these two rational approximations:
The absolute error of the first one is 0.000072… and the absolute error of the second one is 0.000073... so both approximations are (essentially) equally good in terms of accuracy. However, the first one is much simpler to write!
Are there simple, good approximations to any real number? A classical theorem of Dirichlet gives a positive answer.
Theorem (Dirichlet). Let x be a real irrational number. There are infinitely many rational numbers a/b (with a, b coprime integers, say b>0) such that |x-a/b|<1/b².
Can we get even better approximations? not in general. For instance, it is a fun exercise to show that there is no rational approximation a/b to √2 satisfying |√2-a/b|<1/(3·b²).
For a real number x, we let (x) be the distance from x to the integers, that is, the minimum of the quantity |x-n| as n ranges over the integers. Of course, (x) is some number between 0 and 1/2. For instance
where e=2.7182... is Euler’s constant. Another way to look at good simple rational approximations of x is to look for small values that appear early in the sequence
(x), (2x), (3x), (4x), …
Indeed, recall that there is some integer a such that |bx -a|=(bx) and hence
Therefore, we cannot expect (bx) to be significantly smaller than 1/b, as discussed in the previous paragraph. More precisely, the quantity b(bx) can very well be bounded away from 0 as b grows to infinity. For instance, b(b√2)>1/3 for every positive integer b.
Littlewood conjectured that a two variables version should avoid this problem.
Conjecture (Littlewood). For any given real numbers x and y, the sequence n(nx)(ny) takes arbitrarily small values as n grows to infinity.
In [EKL], Einsiedler, Katok and Lindenstrauss used techniques of Ergodic theory to produce an outstanding breakthrough towards Littlewood’s conjecture. They proved that the set of pairs (x,y) for which Littlewood’s conjecture fails is really tiny. Recall from last post that there are 3 steps in most applications of Ergodic theory to number theory: translation into dynamics, classification of measures and equidistribution.
The reduction of Littlewood’s conjecture to a dynamical statement is due to Cassels and Swinnerton-Dyer. Basically, to prove the conjecture for a pair of numbers (x,y) one needs to show that the orbit of certain point in an auxiliary dynamical system X is unbounded. To show unboundedness it would suffice to know that the invariant measures on this dynamical system X are of a very specific form. This classification of invariant measures is still an open question. Einsiedler, Katok and Lindenstrauss succeeded in proving enough of this conjectural classification of invariant measures, so that Littlewood’s conjecture follows in most cases.
[EKL], Einsiedler, Katok, Lindenstrauss. Invariant measures and the set of exceptions to Littlewood’s conjecture. Annals of Mathematics (2006).