Diophantine Approximation is the branch of Number Theory primarily concerned with understanding how well we can approximate real numbers by rationals and variations on this theme. In Metric Diophantine Approximation we are interested in understanding, from various perspectives, the "size" of certain sets of "well-approximable points". In the Euclidean setting, fundamental results of Khintchine and (respectively) Jarnik provide neat characterisations of the Lebesgue measure and Hausdorff measure of the classical sets of well-approximable points. Nowadays there is great interest in extending these results beyond the classical setting in innumerable possible directions. Such extensions draw on ideas and techniques from a variety of areas including, but not limited to, Fractal Geometry, Dynamics, Ergodic Theory, and Number Theory.
If you would be interested in pursuing a PhD in this general area, please get in touch. For a very gentle introduction to the general topic, you may wish to look at the article on pages 11-15 here intended for a general audience which I wrote for the LMS Newsletter.