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Multiplicative zero-one laws and metric number theory

Victor BeresnevichAlan HaynesSanju Velani — 2013

Acta Arithmetica

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...

An application of metric diophantine approximation in hyperbolic space to quadratic forms.

Sanju L. Velani — 1994

Publicacions Matemàtiques

For any real τ, a lim sup set WG,y(τ) of τ-(well)-approximable points is defined for discrete groups G acting on the Poincaré model of hyperbolic space. Here y is a 'distinguished point' on the sphere at infinity whose orbit under G corresponds to the rationals (which can be regarded as the orbit of the point at infinity under the modular group) in the classical theory of diophantine approximation. In this paper the Hausdorff dimension of the set WG,y...

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