Objectives: The main thrust of the proposed research is to develop the metric theory of Diophantine approximation for non-monotonic approximating functions. The research programme naturally splits into two main objectives of equal priority -- the first develops the one dimensional theory while the second develops the higher dimensional theory. Each main objective is comprised of three key aims, which are listed in order of priority.Objective (1) To develop the metric theory of Diophantine approximation on the real line; in particular, to investigate the Duffin--Schaeffer conjecture (Conjecture D-S) -- one of the most difficult and profound unsolved problems in metric number theory. (i) To extend the range of approximating functions for which Conjecture D-S is valid. (ii) To investigate Conjecture D-S for measures close to Lebesgue. measure. (iii) To investigate the Hausdorff measure version of Conjecture D-S for measures significantly different to Lebesgue measure.
Objective (2) To develop the higher dimensional theory of metric Diophantine approximation; in particular, to establish the Hausdorff measure analogues of the Duffin-Schaeffer and Catlin conjectures for linear forms. (i) To extend the Catlin
conjecture to the case of linear forms. (ii) To extend the Duffin--Schaeffer conjecture to the case of linear forms. (iii) To extend the class of approximating functions considered in (i) and (ii) to non-radial approximating functions and thereby obtain a unified linear forms theory.
Final Summary: The project summary provided on the grant proposal is pretty much accurate and the two main objectives were met. In particular, as a direct consequence of the research carried out by the PI and RA we now have a coherent metric theory of Diophantine approximation for non-monotonic approximating functions in higher dimensions. Another significant achievement during the period of the grant was the development of the multiplicative and inhomogeneous theories. The multiplicative theory strengthens the classical results of Gallagher related to the metrical aspect of the famous Littlewood Conjecture.