Objectives at proposal time: The main objective of the proposed research is to make significant contributions to the metric theory of Diophantine approximation on manifolds; in particular to develop a coherent inhomogeneous theory. These are comprised of the following key aims, which are listed in order of priority.i) To establish `inhomogeneous extremality' for non-degenerate manifolds.ii) To establish `inhomogeneous extremality' for friendly measures.iii) To obtain an inhomogeneous Groshev type theory for manifolds.iv) To obtain an inhomogeneous Hausdorff dimension theory for planar curves.v) To obtain a complete inhomogeneous Khintchine type theory for planar curves.
Objectives at report time: As above plus towards the end of the grant the problem of developing a theory of badly approximable points on manifolds was pursued.
Final Summary: The project summary provided on the grant proposal is pretty much accurate and all the key aims were met. In particular, as a direct consequence of the research carried out by the PI, RA and Project Student we now essentially have a ``coherent metric theory of inhomogeneous approximation on manifolds to the same level of understanding as the one of homogeneous approximation''. The idea to develop a ``transfer technique between homogeneous and inhomogeneous extremality'' was spot on and played a key role in establishing the more refined Khintchine-Groshev type results within the inhomogeneous setup. Another significant achievement during the period of the grant was the proof of Schmidt's Conjecture - unsolved for 30 years. The conjecture is concerned with intersecting two of the many various forms of `badly approximable' sets in the plane - on the line there is only one form. At the heart of our proof (involving the PI, RA and Prof Andrew Pollington (NSF, Washington)) lies the very simply idea of fixing a vertical line in the plane and understanding the manner in which any given badly approximable set intersects the fixed line. At first glance this may seem unconnected to main thrust of the proposed research; i.e. approximation on manifolds. However, this is far from the truth since a vertical line is a manifold (albeit a simple one) and we are investigating a special case of the ``badly approximable'' analogue of the Baker-Sprindzuk conjecture. The problem of finding uncountably many badly approxiamble points on a planar curve is explicitly stated in a paper of Davenport from the fifties. Currently, the PI and RA (who now has a permanent position at Durham) are investigating this problem of Daveport. In short the hope is to modify the proof of Schmidt in a manner that allows us to replace vertical lines by curves. This ongoing work and indeed the proof of Schmidt arose principally as a result of the grant.