Metrical Diophantine approximation for quaternions

Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.Comment: 30 pages. Some minor improvement

Classical metric Diophantine approximation revisited

The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored.Comment: 31 pages, Dedicated to Klaus Roth on the occasion of his 80th birthda

A Markov Chain based method for generating long-range dependence

This paper describes a model for generating time series which exhibit the statistical phenomenon known as long-range dependence (LRD). A Markov Modulated Process based upon an infinite Markov chain is described. The work described is motivated by applications in telecommunications where LRD is a known property of time-series measured on the internet. The process can generate a time series exhibiting LRD with known parameters and is particularly suitable for modelling internet traffic since the time series is in terms of ones and zeros which can be interpreted as data packets and inter-packet gaps. The method is extremely simple computationally and analytically and could prove more tractable than other methods described in the literatureComment: 8 pages, 2 figure

On the entropy of curves

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