Classical metric Diophantine approximation revisited

Abstract

The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored.

31 pages, Dedicated to Klaus Roth on the occasion of his 80th birthday

References in corpus (1)

• Ubiquity and a general logarithm law for geodesics

Cited by in corpus (11)

• A note on zero-one laws in metrical Diophantine approximation
• A note on the Duffin-Schaeffer conjecture with slow divergence
• On shrinking targets for Z^m actions on tori
• The Divergence Borel-Cantelli Lemma revisited
• On parametric Thue-Morse Sequences and Lacunary Trigonometric Products
• Zero-one laws in simultaneous and multiplicative Diophantine approximation
• Long hitting time, slow decay of correlations and arithmetical properties
• Quantitative non-divergence and Diophantine approximation on manifolds
• Khintchine's theorem and Diophantine approximation on manifolds
• Small Furstenberg sets
• Badly approximable affine forms and Schmidt games