M. M. Dodson

Diophantine approximation and a lower bound for Hausdorff dimension

Sets of the general form where U is a subset of ℛ k and is a family of subsets of U indexed by a set J , are common in the theory of Diophantine approximation [4, 7, 18, 19]. They are also closely connected with exceptional sets arising in analysis and with sets of “small divisors” in dynamical systems [1, 8, 15”. When J is the set of positive integers ℕ, the set Λ(ℱ) is of course the lim-sup of the sequence of sets F j , j = 1, 2,… [11, p. 1]. We will also call sets of the form (1), with the more general index set J, lim-sup sets. When such lim-sup sets have Lebesgue measure zero, it is of interest to determine their Hausdorff dimension. It is usually difficult to obtain a good lower bound for the Hausdorff dimension (and it can be much harder to determine than an upper bound). In this paper we will obtain a lower bound for the dimension of lim-sup sets of the form (1) for a fairly general class of families ℕ which includes a range of results in the theory of Diophantine approximation. This lower bound depends explicitly on the geometric structure and distribution in U of the sets F α in ℕ.

Number theory and dynamical systems

1. Non-degeneracy in the perturbation theory of integrable dynamical systems Helmut Riissmann 2. Infinite dimensional inverse function theorems and small divisors J. A. G. Vickers 3. Metric Diophantine approximation of quadratic forms S. J. Patterson 4. Symbolic dynamics and Diophantine equations Caroline Series 5. On badly approximable numbers, Schmidt games and bounded orbits of flows S. G. Dani 6. Estimates for Fourier coefficients of cusp forms S. Raghavan and R. Weissauer 7. The integral geometry of fractals K. J. Falconer 8. Geometry of algebraic continued fractals J. Harrison 9. Chaos implies confusion Michel Mendes France 10. The Riemann hypothesis and the Hamiltonian of a quantum mechanical system J. V. Armitage.

Metric Diophantine approximation and Hausdorff dimension on manifolds

In this paper we discuss homogeneous Diophantine approximation of points on smooth manifolds M in ℝ k . We begin with a brief survey of the notation and results. For any x,y ∈ℝ k , let .

Extremal manifolds and Hausdorff dimension

The recent proof by D. Y. Kleinbock and G. A. Margulis [11] of Sprindžuk’s conjecture for smooth nondegenerate manifolds M means that the set Lv(M) of v-approximable points (this and other terminology is explained below) on M is of zero induced Lebesgue measure. This raises the question of its Hausdorff dimension.

Homogeneous additive congruences

An investigation of conditions under which the congruence a1xk 1 + ... + aa sXks =0 (mod pn), where a1, ..., a5 are any non - zero integers and pn is any prime power, has a primitive solution.

Darwin's law of natural selection and Thom's theory of catastrophes

Abstract The fitness of a population is defined to be a real smooth function of its environment and phenotype. Darwins law of natural selection implies that a population in equilibrium with its environment under natural selection will have a phenotype which maximizes the fitness locally. By using Thom theory it is possible under a number of assumptions to make qualitative inferences about the phenotypic change of a population subject to natural selection in a continuously varying environment. When the environment is one or two dimensional, Thoms catastrophe theorem implies that sudden and substantial changes in phenotype can only arise from the fold and cusp catastrophes. It is shown that a population in stable equilibrium with its environment exhibits genetic assimilation, and that the main modes of evolution can arise from appropriate variations in the environment.

Simultaneous Diophantine approximation on the circle and Hausdorff dimension

The functional relations between the coordinates of points on a manifold make the study of Diophantine approximation on manifolds much harder than the classical theory in which the variables are independent. Nevertheless there has been considerable progress in the metric theory of Diophantine approximation on smooth manifolds. To describe this, some notation and terminology are needed.

On the approximate form of Kluvánek's theorem

An abstract form of the classical approximate sampling theorem is proved for functions on a locally compact abelian group that are continuous, square-integrable and have integrable Fourier transforms. An additional hypothesis that the samples of the function are square-summable is needed to ensure the convergence of the sampling series. As well as establishing the representation of the function as a sampling series plus a remainder term, an asymptotic formula is obtained under mild additional restrictions on the group. In conclusion a converse to Kluvaneks theorem is established.

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 on 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 modeling Internet traffic because the time series is in terms of ones and zeros, which can be interpreted as data packets and interpacket gaps. The method is extremely simple, both computationally and analytically, and could prove more tractable than other methods described in the literature.

Dirichlet's theorem and diophantine approximation on manifolds

We show that if M ⊂ Rk belongs to a general class of smooth manifolds then, for almost all x ∈ M, Dirichlets theorem on Diophantine approximation cannot be infinitely improved.