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Metric Number Theory and Diophantine approximation. Other research interests: geometry of numbers, uniform distribution, measure and probability theory, fractal geometry, ergodic theory, dynamical systems, applications of Diophantine approximation (in PDEs, signal processing, etc.).
Victor Beresnevich, Jason Levesley and Sanju Velani work on a variety of problems in metric number theory and Diophantine approximation that involve a range of techniques from Diophantine approximation, analytic number theory, the geometry of numbers, probability theory, fractals and ergodic theory. Some examples include the Duffin-Schaeffer problem on rational approximations to real numbers, problems on approximation by algebraic numbers, problems on badly approximable vectors, problems on Diophantine approximation on manifolds, etc. The Diophantine approximation problems have natural `dynamical' analogues in terms of shrinking targets problems associated with the phase space of a given dynamical system. Victor Beresnevich and Sanju Velani are currently running a large scale research programme and any PhD student would become an integral part of the larger research group. If interested, please, contact either of them for possible PhD research projects.