New frameworks in metric Number Theory: foundations and applications

Diophantine approximation is a branch of number theory that can loosely be described as a quantitative analysis of the property that every real number can be approximated by a rational number arbitrarily closely; i.e. the rationals are dense in the real line. The theory dates back to the ancient Greeks and Chinese who used good rational approximations to the number pi (3.14...) in order to accurately predict the position of planets and stars. Today, the theory is deeply intertwined with other areas of mathematics such as ergodic theory, dynamical systems and fractal geometry. It continues to play a significant role in applications to real world problems including those arising from the rapidly developing areas of electronic communications, antenna design and signal processing.

While yielding spectacular achievements over centuries, the development of the theory of Diophantine approximation has crystallised some of today's major research challenges in mathematics; in particular the conjectures of Littlewood and Duffin-Schaeffer and the generalised Baker-Schmidt problem. These challenges form the backbone of the research programme. In short, we plan to develop bold new frameworks in metric Diophantine approximation with the goal of solving challenging and topical problems.

The metrical theory of Diophantine approximation is the study of the approximation properties of real numbers by rationals from a measure theoretic (probabilistic) point of view. The central theme of this theory is to determine whether a given approximation property holds everywhere except on an exceptional set of measure zero. Littlewood's Conjecture, which predicts how well pairs of real numbers can be multiplicatively approximated by rationals with the same denominator, is a universal statement in that the associated approximating property is required to hold for all points. Transforming universal Diophantine approximation problems into 'twisted' probabilistic problems is an example of one the novel frameworks to be developed. In essence this would allow us to use the language and machinery developed in metrical Diophantine approximation for problems that a priori are not connected with metrical number theory. This novel reformulation, then, would allow a fresh attack on some long-standing conjectures. For instance, in the case of Littlewood's Conjecture, the probabilistic reformulation arises naturally by 'twisting' the standard (inhomogeneous) theory of metrical Diophantine approximation. Even this represents unexplored territory. In mathematics - and indeed in science - a new viewpoint can be the key to solving an old problem and moreover can lead to flourishing new theories.

In the past the three aforementioned research challenges were thought to involve disparate ideas. However recent advances have shown that there are substantial links between them. For example, fixing one of the real numbers in Littlewood's Conjecture enables us to recast the problem in terms of the one-dimensional setting of the Duffin-Schaeffer Conjecture (the error of approximation in non-monotonic). In turn, by making fundamental use of the Ostrowski numeration of numbers this has lead to new metrical insights into Littlewood's Conjecture. Also, restricting a two-dimensional approximation problem to a line (or more generally a curve) naturally brings into play the theory of Diophantine approximation of manifolds. The generalised Baker-Schmidt problem is central to the development of this theory and is intimately linked to key problems regarding the distribution of rational points near manifolds. The exploitation of the links between the research challenges is a key feature of the programme.

ENHANCING THE UK'S INTERNATIONAL STANDING
This programme is ambitious. Significant progress with any of the research challenges will be a major success. The programme will allow us to develop exciting new directions that will greatly influence the broader area of Diophantine approximation for many years to come -- see `Academic Beneficiaries' for details. Four decades ago the UK became a world leader in this area with dazzling contributions being made by eminent mathematicians including Klaus Roth and Alan Baker who were awarded Fields Medals (the mathematical equivalent of the Nobel Prize). The novel nature of the proposed research will build upon the Diophantine approximation tradition in the UK and enhance the UK's leadership in the field.

ENHANCING PEOPLE PIPELINE AND DELIVERING HIGHLY SKILLED RESEARCHERS
Postdoctoral positions are vital to the continued success of the UK on the world stage. They allow talented researchers to develop the skills necessary to become the next generation's leaders. The importance of maintaining the UK's people pipeline was highlighted at the International Review of Mathematical Sciences held in Durham last year. Both the PI and CI attended this: the PI was invited to represent Number Theory research in the North-East. The main investment within our programme will be used to create six 3-year postdoctoral positions. While the posts will be offered to the best candidates, we seek to hire the UK's most promising early career researchers. This will be a vital factor in helping ensure that outstanding researchers stay in the country and launch successful long-term careers in the UK. Given the length of the grant then we hope that these benefits will also extend to encouraging gifted undergraduates to stay on and pursue a career in science as they will be made aware of the existence of exciting job opportunities. The PDRAs will have many opportunities to develop their intellectual and transferable skills. This way the programme will deliver highly skilled researchers able to benefit both the scientific and technological communities and strengthen the UK's mathematics research base. This underpins science and engineering at all levels.

IMPACT ON ECONOMY AND SOCIETY
Number Theory is known to have major impact on applied disciplines. For example, in dynamical systems, analysing resonance phenomena leads to the notorious problem of small denominators that can be dealt with using Diophantine approximation. A wide range of applied problems can be treated this way, most famously using the Kolmogorov-Arnol'd-Moser theory. Definite and topical applications of this kind are currently evolving in the rapidly developing areas of electronic communication, antenna design and signal processing, which (surprisingly) make use of the full power of recent advances in the theory of Diophantine approximation particularly the work of the investigators. For example, the papers [arXiv:0908.2282], [arXiv:0908.1208], [arXiv:1101.2182] exploit the CI's work on the Khintchine-Groshev theory for manifolds in order to investigate the potential of multiple-input and multiple-output (MIMO) technologies. These technologies have attracted attention in wireless communications for their high efficiency and reliability and are having a significant impact on modern mobile communication standards. Mobile communications change fast with consumer demand growing. Wireless technologies are used in all walks of life including computing, media and business. The fundamental research addressed within this programme will enhance any development process involving ideas from number theory, adding to the store of knowledge that underpins technology. The scope is potentially huge, with savings made in the communication equipment and more efficient energy factors. This contributes both to wealth creation and the improvement of quality of life.