Multiplicative zero-one laws and metric number theory

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.

Metadata

Authors/Creators:

Beresnevich, Victor https://orcid.org/0000-0002-1811-9697
Haynes, A https://orcid.org/0000-0001-6077-8162
Velani, Sanju https://orcid.org/0000-0002-4442-6316

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Keywords:

Diophantine approximation, zero-one laws, Duffin-Schaeffer conjecture

Dates: