Circle rotations and their generalisations in Diophantine approximation

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.

1) Increasing security of internet transactions: The accomplishment of milestone (D2i) in my Case for Support will increase global knowledge of the security of the RSA encryption algorithm. This is currently the most widely used algorithm for public encryption of data, and most internet transactions and signatures rely on this algorithm to ensure their validity. When we share information with a secure web site or disclose our credit card and personal information online we rely on the security of the RSA algorithm to protect this information from unauthorized parties.

On the negative side there is a global proliferation of online phishing attacks, in which internet criminals fraudulently disguise themselves as banks and other legitimate organizations in order to steal personal data from unsuspecting victims. This has resulted in massive losses, with losses from online bank fraud alone amounting to tens of millions of pounds each year (according to recent conservative estimates by Microsoft). RSA encryption is one of the safeguards that banks and other institutions currently use in order to establish and verify secure connections to minimize the damage caused by phishing. This research proposal will help us to understand more of the possible weaknesses of the RSA algorithm, which is crucial to closing the door on the illegal compromise of personal data.

RSA encryption is always a topic of major interest in computer science. As such publishing our results online and in a major mathematical journal will ensure that researchers and people who are applying these algorithms to daily life will be aware of our advances. Furthermore the University of Bristol is closely connected with the Heilbronn Institute, which will also aid in quickly disseminating our findings to the correct parties.

2) Strengthening the people pipeline in the UK: I attended a recent International Review of Mathematics meeting in Durham, where it was pointed out strongly that one of the key problems facing mathematics in the UK is the insufficiency of post-doctoral positions for recent PhDs. Related to this need, one of the provisions of this research proposal is for the appointment of an RA for 3 years. This will help to strengthen the people pipeline in the UK and it will provide a solid foundation for the RA to establish a successful career in mathematics.

This project will have a considerable academic impact on the RA, who is to join me for three years beginning in the second year of the grant. My goal is to select and train a promising recent PhD student coming from either number theory or dynamical systems, in order both to extend the RA's mathematical knowledge base and to help them to establish international connections with other mathematicians working in related fields. Ideally this would be a person from the UK, and I have several people in mind. Being involved in these widely respected and far reaching problems at the forefront of current research will open many doors and prepare the RA for a successful career in mathematics.

3) Helping to maintain the diversity and international standing of the UK as a whole: Part of the proposal includes plans to travel worldwide and to invite people to the UK, which will help to maintain the diversity and international links that the UK has with other parts of the world. Also if certain objectives of the proposal are successful, for example if we succeed in proving the Littlewood Conjecture, it is likely to draw international media attention because of the growing interest in mathematics among the general public.