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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-2-1,
author = {Victor Beresnevich and Alan Haynes and Sanju Velani},
title = {Multiplicative zero-one laws and metric number theory},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {101-114},
zbl = {1292.11085},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-2-1}
}
Victor Beresnevich; Alan Haynes; Sanju Velani. Multiplicative zero-one laws and metric number theory. Acta Arithmetica, Tome 161 (2013) pp. 101-114. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-2-1/