Diophantine approximation: the twisted, weighted and mixed theories

This PhD thesis consists of five papers dealing with problems in various branches of Diophantine approximation. The results obtained contribute to the theory of twisted, weighted, multiplicative and mixed approximation.

In Paper I a twisted analogue of the classical set of badly approximable linear forms is introduced. We prove that its intersection with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

The main result of Paper II concerns a weighted version of the classical set of badly approximable pairs. We establish a new characterization of this set in terms of vectors that are well approximable in the twisted sense. This naturally generalizes a classical result of Kurzweil. In Paper II we also study the metrical theory associated with a weighted variant of the set introduced in Paper I. In particular, we provide a sufficient condition for this variant to have full Hausdorff dimension. This result is extended in Paper III to imply the stronger property of 'winning'.

Paper IV addresses various problems associated with the Mixed Littlewood Conjecture. Firstly, we solve a version of the conjecture for the case of one p-adic value and one pseudo-absolute value with bounded ratios. Secondly, we deduce the answer to a related metric question concerning numbers that are well approximable in the mixed multiplicative sense. This provides a mixed analogue to a classical theorem of Gallagher.

In Paper V we develop the metric theory associated with the mixed Schmidt
Conjecture. In particular, a Khintchine-type criterion for the 'size' of the natural set
of mixed well approximable numbers is established. As a consequence we obtain a
combined mixed and weighted version of the classical Jarník-Besicovich Theorem.