Circle rotations and their generalisations in Diophantine approximation

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Diophantine approximation is the study of how well real numbers can be approximated by rational numbers. Throughout the history of mathematics this has been one of the most important fields in applications to real world problems. Today Diophantine approximation is used in numerical algorithms and computer programs which model scientific experiments and other natural behaviour. It also plays a significant role as a supporting structure for results in many other mathematical and scientific settings.There are several long standing open problems in Diophantine approximation which have attracted recent attention in the wider mathematical community. One of these is the Littlewood Conjecture, which predicts how well pairs of real numbers can be simultaneously approximated by rationals with the same denominator. The goal of this project is to investigate the Littlewood Conjecture and related problems by using information about the distribution of circle rotations and their generalisations.Suppose you take a circle of circumference one and single out a point somewhere along the boundary. If you rotate the whole circle through a fixed angle your point will move to a new position on the circle. If you think about repeating this rotation infinitely many times then the collection of all possible positions of the point is called its orbit. Understanding the orbits of points under a given rotation is a basic problem which is directly related to understanding how well a real number can be approximated by fractions.I have recently shown how a technique called Ostrowski expansion can be used to prove substantial new results about the Littlewood Conjecture. Ostrowski expansion basically allows us to reorganize the orbits of points into an infinite array of blocks, each of which can then be understood by using number theoretic techniques. In this way the Ostrowski expansion can be used to isolate one of the variables in the Littlewood Conjecture and thereby recast the problem in a one-dimensional setting.This understanding of circle rotations may well lead to the proof of the entire Littlewood Conjecture. However there are also several other interesting problems which are open to attack via this method.One such problem which I will investigate is known as the "shrinking targets" problem. Here you consider a circle rotation and to each element in the orbit of a point you attach a small ball of a certain radius. The radii of the balls should shrink as the rotation progresses, and the problem is to determine which points on the circle are captured in infinitely many of the balls. In the form presented here the answer to this problem is known. However it is still a wide open problem to prove a quantitative result, which would tell us something about the proportion of balls which capture a given point on the circle. These types of problems have consequences in dynamical systems and particle physics.Another problem is to replace the circle rotation by a different transformation of the circle. The so-called "interval exchange transformations" are generalisations of circle rotations which are relevant to problems in Diophantine approximation and dynamical systems. It is possible to associate to each of these transformations an Ostrowski expansion that encodes information about the orbits of points. In this way the framework which we are developing to study the Littlewood Conjecture should also allow us to prove new and interesting results in many settings.