Multiplicative zero-one laws and metric number theory

TY - JOUR

T1 - Multiplicative zero-one laws and metric number theory

AU - Beresnevich, Victor

AU - Haynes, A

AU - Velani, Sanju

N1 - This is an author-produced version of a paper accepted for publication. Uploaded with permission of the publisher/copyright holder. Further copying may not be permitted; contact the publisher for details

PY - 2013/6

Y1 - 2013/6

N2 - 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.

AB - 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.

KW - Diophantine approximation

KW - zero-one laws

KW - Duffin-Schaeffer conjecture

U2 - 10.4064/aa160-2-1

DO - 10.4064/aa160-2-1

M3 - Article

SN - 0065-1036

VL - 160

SP - 101

EP - 114

JO - Acta Arithmetica

JF - Acta Arithmetica

IS - 2

ER -