Multiplicative zero-one laws and metric number theory

TL;DR

This paper extends classical Diophantine approximation theory by establishing a multiplicative zero-one law and an analogue of the Duffin-Schaeffer theorem, using a versatile cross fibering principle to handle higher-dimensional cases.

Contribution

It introduces a multiplicative zero-one law and an analogue of the Duffin-Schaeffer theorem without monotonicity or convexity assumptions, using the cross fibering principle.

Findings

Established a complete multiplicative zero-one law.

Proved an analogue of the Duffin-Schaeffer theorem in the multiplicative setting.

Developed the cross fibering principle for higher-dimensional lifting.

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.