Article

Multiplicative zero-one laws and metric number theory

12/2010;
Source: arXiv

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.

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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;
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    ABSTRACT: This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ if and only if $\sum_n\lambda({\mathcal E}_n)=\infty$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, \[ {\mathcal E}_n={\mathcal E}_n(\psi)=\bigcup_{m=1 \atop (m,n)=1}^n\big(\frac{m-\psi(n)}{n},\frac{m+\psi(n)}{n}\big), \] $\psi$ is any non-negative arithmetical function. Instead of studying $\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n$ we introduce an even fundamental object $\cup_{n=1}^{\infty}{\mathcal E}_n$ and conjecture there exists a universal constant $C>0$ such that \[\lambda(\bigcup_{n=1}^{\infty}{\mathcal E}_n)\geq C\min\{\sum_{n=1}^{\infty}\lambda({\mathcal E}_n),1\}.\] It is shown that this conjecture is equivalent to the Duffin-Schaeffer conjecture. Similar phenomena are found in the fields of $p$-adic numbers and formal Laurent series. As a byproduct, we answer conditionally a question of Haynes by showing that one can always use the quasi-independence on average method to deduce $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ as long as the Duffin-Schaeffer conjecture is true. We also show among several others that two conjectures of Haynes, Pollington and Velani are equivalent to the Duffin-Schaeffer conjecture, and introduce for the first time a weighted version of the second Borel-Cantelli lemma to the study of the Duffin-Schaeffer conjecture.
    12/2013;

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