Effective Equidistribution in Diophantine Approximation : Theory, Interactions 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. The theory dates back to the ancient Greeks and Chinese who used good 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 many other areas of mathematics such as ergodic theory, dynamical systems, probability theory and fractal geometry. It also continues to play a significant role in applications to real world problems including those arising from computer science or from the rapidly developing areas of electronic communications, antenna design and signal processing.

The many interactions between Diophantine Approximation and other disciplines in science can be explained by the universal need to approximate complex structures by more regular ones. Many problems can thus be reduced to the analysis of the distribution of sets of approximation. They can furthermore be solved upon having a good enough understanding of the regularity such sets enjoy (that is, their equidistribution properties). The research project aims at exploiting this fruitful point of view to tackle some deep and long-standing problems lying at heart of topics as varied as Metric (i.e. probabilistic) Number Theory, Convex Geometry and Diophantine Analysis.

One of the goals is related to the problem of approximating dependent quantities (e.g., a number and its cube) by rationals. This leads to the domain of Diophantine Approximation on manifolds, where for most questions no general theory is available. Our aim is to develop such a general theory for a large class of curves by determining the fractal dimension of very well approximable points lying on them. This is related to the problem of counting the number of rational points with bounded denominators lying close to the given curves and constitutes an extremely active domain of research.

The project will also be concerned with a question in Convex Geometry which can loosely (but surprisingly simply) be described as follows: suppose you stand in a forest where all tree trunks have the same (very small) size. Can you position them in such a way that they are at least a unit distance apart from each other and that no matter where you stand and what direction you look in, you will never be able to see the horizon? If so, what is the smallest visibility that can be guaranteed upon arranging the trees in a suitable way? The underlying deep problem is due to Danzer (1965) and is closely related to other questions in mathematical physics and in the theory of mathematical quasicrystals. It is still open, and will be addressed with Diophantine methods by analysing the distribution of the set of points defined as the centers of the trees.

Finally, another goal will be to answer some questions related to problems of effectivity (that is, to problems where it is known that a result holds, but where it is not known how to check it in any concrete example). More precisely, the focus will be on sequences of numbers satisfying a simple property such as the following: whenever, say, three consecutive ones are known, then the number coming after them can be deduced from a fixed simple rule such as adding the three given numbers and multiplying them by constants. It should be clear that the data of the first three numbers in the list and of the (fixed) rule of deduction of the fourth one determines the entire list. It can then be shown that, under favorable conditions, the number of zeros appearing in this list is always finite. The question, at the heart of deep problems of decidability in Computer Science for instance, is to be able to find a range beyond which it can be guaranteed that all terms are nonzero. This question is studied as part of the proposed project.

The goal of this project is to exploit and enhance the interactions between Diophantine Approximation and other areas of mathematics in order to make significant advances in long-standing open problems related to Metric Number Theory, Convex Geometry and Diophantine Analysis. The proposed research project is therefore naturally intradisciplinary, although some of the problems it aims at solving stem from questions posed by Computer Scientists and are of interest to specialists working in Numerical Analysis (for instance). In this respect, it must be underlined that the PI has been actively contributing to the interaction between pure sciences and applied disciplines through ongoing collaborations with engineers. As he has also been involved in outreach activities that he intends to relate to the problems which are part of the proposal, the project is expected to have a far-reaching and a wide-ranging impact.

The theoretical part is abstract by nature and lies at the cutting edge of research in Diophantine Approximation, whose utility to other disciplines in science is well-established (see the Academic Beneficiaries section). Since the proposed research impacts and/or draws upon ideas from many areas (e.g., Probability Theory, Geometry and Mathematical Physics), it will contribute to fostering a transfer of knowledge between various disciplines within mathematics in order to provide new bridges or to enforce the existing ones. Considering that the aim is to tackle long-standing open problems, it is expected that the research programme will add to the existing knowledge and expertise notably, but not only, in Number Theory in the UK. The various collaborations which have been started with leading scientists will also contribute to strengthening the existing links with the mathematical community at an international level. Also, the talks the PI will deliver, the high-quality research papers he intends to publish and the surveys he will write up on problems which deserve to be better known within the scientific community will, too, be very profitable for research in the UK.

Although the main theme of the research programme lies at the heart of pure mathematics, some of the problems it is concerned with are well-known to have implications well beyond the field of pure science, and ultimately in "real life" problems. This includes view obstruction problems (studied in Computer Graphics), problems of decidability (studied in Computer Science) and the search for optimal quadrature formulae (studied in Numerical Analysis). Also, deep and fundamental results in Diophantine Approximation have recently found unexpected and far-reaching applications in the rapidly developing areas of electronic communication, antenna design and signal processing. This transfer of knowledge from pure mathematics to engineering science is an ongoing and very active process relying on unexpected applications of theoretical results to very concrete problems. It is therefore likely that the technological communities will also benefit from the theories developed in this project in the middle term.

More generally, since the project would contribute to the UK's mathematics research base, which underpins science and engineering, the scientific and technological communities would benefit from it.

As for the wider public, the PI will popularize the aspects of pure mathematics part of the project by giving talks in informal forums such as 'Cafés scientifiques' and by meeting classes of students in high-schools. This is undoubtedly the best way to get people interested in mathematics, to attract students to science and to inspire the next generation of researchers.