New frameworks in metric number theory: foundations and applications

The Engineering and Physical Sciences Research Council (EPSRC*) has awarded a £1.6 million Programme Grant (£2 million fEC) to Professors Sanju Velani and Victor Beresnevich at the Department of Mathematics of the University of York. Starting 1 June 2012 this six years award will fund a wide range of activities and appointments including 6 Research Assistantships, short and long term research visitors and an international workshop.

The research programme aims to make significant contribution to fundamental problems in the theory of Diophantine approximation. While yielding spectacular achievements over centuries, the development of Diophantine approximation has crystallised some of today's major research challenges; in particular Littlewood's Conjecture (1930) (PDF , 82kb) on multiplicative Diophantine approximation, the Duffin-Schaeffer (PDF , 89kb) on non-monotonic Diophantine approximation and the generalised Baker-Schmidt (PDF , 108kb) on Diophantine approximation on manifolds. These challenges form the backbone of our research programme.   

Diophantine approximation** continues to have important and exciting applications in other research disciplines and indeed real world problems. Definite and topical applications of Diophantine approximation are currently evolving in the rapidly developing area of electronic communication, which make use of the full power of recent advances in the theory of metric Diophantine approximation, in particular, work on the Khintchine-Groshev theory for manifolds in order to investigate the potential of multiple-input and multiple-output (MIMO) technologies. An inherent part of this programme will be to contribute to such applications and enhance any development process arising from the real world problems and involving ideas from number theory. 

*EPSRC is the main UK government agency for funding research and training in engineering and the physical sciences. "EPSRC Programme Grants are a flexible mechanism to provide funding to world-leading research groups to address significant major research challenges."  Prior to York there were only 3 other programme grants across mathematical sciences awarded to run a research-led project, namely those held at Cambridge, Imperial and Oxford.

**Diophantine approximation is a branch of number theory that can loosely be described as a quantitative analysis of the property that the rationals are dense in the real line. The theory dates back to the ancient Greeks and Chinese who discovered good rational approximations to π in order to predict the position of planets and stars. Nowadays the theory is deeply intertwined with measure theory, ergodic theory, dynamical systems, fractal geometry and other areas of mathematics.