Multiplicative zero-one laws and metric number theory
Volume 160 / 2013
Acta Arithmetica 160 (2013), 101-114
MSC: 11J13, 11J83, 11K60.
DOI: 10.4064/aa160-2-1
Abstract
We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin–Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile `cross fibering principle'. In a nutshell it enables us to `lift' zero-one laws to higher dimensions.
