Effective Equidistribution in Diophantine Approximation : Theory, Interactions and Applications.

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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.