Metric Diophantine approximation — from continued fractions to fractals

The metric theory of Diophantine approximation dates back to Èmile Borel (1909) and Khintchine (1924). Starting from Khintchine, we will give a survey of metric Diophantine approximation, with tthe occurrence of fractal structures in number theory playing a major part. Particular emphasis will be put on recent efforts to tackle major outstanding problems via two different avenues: One uses classical techniques from fractal geometry and number theory, while the other uses methods from homogeneous dynamics.