Theory of badly approximable sets

A given real number can be approximated by rational numbers with an arbitrary precision. For example the decimal representation of a number provides such an approximation. On the other hand different numbers are approximated by rationals with different speed in terms of the denominators of that rational numbers. There is no upper bound for that speed which was firstly mentioned by Liouville in 1840's. On the other hand there is a natural lower bound for that speed which is determined by Dirichlet-Hurwitz theorems. It says that for any irrational number x one can find infinitely many rational numbers p/q such that

|x - p/q| < 1/q^2.

Moreover there exist irrational numbers which make this inequality sharp. In other words for that numbers the distance |x-p/q| can not be made smaller than c/q^2 for some positive constant c and all rational numbers p/q. Irrational real numbers with such property are called badly approximable.

The set of badly approximable numbers can be described quite well with help of the theory of continued fractions. It allows us to derive a lot of information about that set: it's "size" in terms of Lebesgue's measure and Hausdorff dimension and some other information about its structure.

In two-dimensional case and more generally in high dimensional case we can also approximate any point in R^n by points with rational coefficients. However different coordinates can be approximated by rationals with their own speed. Because of this phenomena, in higher dimensions we have an uncountable family of different sets of badly approximable points depending on the relative speed of approximation of different coordinates of the point.

The structure of the sets of badly approximable points in high dimensions is much less known than their one-dimensional analogue. It incorporates a lot of open problems. Some of them like famous Littlewood conjecture attract a lot of attention of the modern mathematical society. In this project I am going to shed the light on many of that problems.

One of the main objectives of this project is to construct the mechanism which, like continued fractions in one dimension, will help us to deal with badly approximable points easier. This work has already been started by me together with Velani and Pollington. We described the sets which allowed us to solve a number of related problems including the famous Schmidt conjecture.

Next, the mechanism of generalized Cantor sets will be used to solve some particular open problems about badly approximable points in n-dimensional real space. In particular the structure of BAD points on manifolds in arbitrary dimensions will be investigated.

Finally some new approaches will be applied to attack the famous Littlewood conjecture. It was stated by Littlewood in 1931 and nowadays it is one of the most attractive problems in modern Mathematics.

The research proposed in this application primarily aims at developing new techniques and solving outstanding problems in the theory of Diophantine approximation. For decades this area of
research has been a UK strength and has led to numerous breakthroughs, some of them being acknowledged by Fields medals (Roth, Baker). Nowadays the UK position in Diophantine approximation as well as in other research areas particularly close to the proposed research such that ergodic theory and dynamical systems are strong and represented across the country, e.g. at Bristol, Manchester, Warwick, York and other places. Doubtless the UK will continue to hold one of the world-leading positions in the area. The proposed research will help maintaining the UK leadership and contribute to the development of its research capability in this research area. This will highly likely remain important over the next 10-50 years as the various high-profile research activities in the area have only recently started unfolding.

The proposed topics of the research are also very well recognized by the international mathematical society. So the mathematical society on both national and international levels will certainly benefit from this project.