Multiplicative zero-one laws and metric number theory

Acta Arithmetica (Impact Factor: 0.42). 12/2010; 160(2). DOI: 10.4064/aa160-2-1
Source: arXiv


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.

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Available from: Victor Beresnevich, May 27, 2015
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    • "The extra log factor in the above sum accounts for the larger volume of the fundamental domains defined by (4) compared to (3). The recent work [9] has made an attempt to relax the monotonicity assumption on ψ within the multiplicative setting. Our goal in this paper is to investigate the Hausdorff measure theory within the multiplicative setting. "
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    ABSTRACT: The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider three inhomogeneous problems that further develop these classical results. Firstly, we obtain a Jarnik type theorem for the set of multiplicatively approximable points in the plane. This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting. Next, we establish a Jarnik type theorem for the set of multiplicatively approximable points on a non-degenerate planar curve. This completes the Hausdorff theory for planar curves. Finally, we show that the set of simultaneously inhomogeneously (i,j)-badly approximable points in the plane is of full dimension. The underlying philosophy behind the proof has other applications; e.g. towards establishing the inhomogeneous version of Schmidt's Conjecture. The higher dimensional analogues of the planar results are also discussed.
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    • ". For the case G ∈ {A, B} & n = 1 in multiplicative Diophantine approximation, we refer to [4] for appropriate conjectures and [4] [12] for Duffin-Schaeffer and Khintchine-Groshev types theorems. For the case G = C & n = 1 & m ≥ 2, we would like to propose the following conjecture. "
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    ABSTRACT: Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani.
    04/2012; 59(2). DOI:10.1112/S0025579312001106
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    ABSTRACT: We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric theory associated with mixed problems in this setting. This includes finding a natural mixed analogue of the classical Jarn\'ik-Besicovich Theorem. Previous knowledge surrounding mixed problems was almost entirely restricted to the multiplicative setup of de Mathan & Teuli\'e [21], where the concept originated.
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