The classical theory of Diophantine approximation quantifies the density of rational number in the real line. In a joint work with A. Ghosh and A. Nevo we consider an analogous problem of approximating by dense orbits for actions on homogeneous spaces. We explain a general approach which allows to establish quantitative density and gives the best possible exponents of approximation in a number of cases.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/5180
[4] https://www.mpim-bonn.mpg.de/node/5079