Rational approximation and arithmetic progressions

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed

Ultrametric Logarithm Laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock-Margulis and obtain related results in Metric Diophantine Approximation.Comment: submitted to Montasefte Fur Mathemati

Higher-rank Bohr sets and multiplicative diophantine approximation

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.Comment: arXiv admin note: text overlap with arXiv:1703.0701

On Kronecker's density theorem, primitive points and orbits of matrices

We discuss recent quantitative results in connexion with Kronecker's theorem on the density of subgroups in R^n and with Dani and Raghavan's theorem on the density of orbits in the spaces of frames. We also propose several related problems. The case of the natural linear action of the unimodular group SL_2(Z) on the real plane is investigated more closely. We then establish an intriguing link between the configuration of (discrete) orbits of primitive points and the rate of density of dense orbits

On the Wiener-Hopf factorization for Levy processes with bounded positive jumps

We study the Wiener-Hopf factorization for Levy processes with bounded positive jumps and arbitrary negative jumps. Using the results from the theory of entire functions of Cartwright class we prove that the positive Wiener-Hopf factor can be expressed as an infinite product in terms of the solutions to the equation $\psi(z)=q$, where $\psi$ is the Laplace exponent of the process. Under some additional regularity assumptions on the Levy measure we obtain an asymptotic expression for these solutions, which is important for numerical computations. In the case when the process is spectrally negative with bounded jumps, we derive a series representation for the scale function in terms of the solutions to the equation $\psi(z)=q$. To illustrate possible applications we discuss the implementation of numerical algorithms and present the results of several numerical experiments.Comment: 29 pages, 4 figure

A Contribution to Metric Diophantine Approximation : the Lebesgue and Hausdorff Theories

This thesis is concerned with the theory of Diophantine approximation from the point of view of measure theory. After the prolegomena which conclude with a number of conjectures set to understand better the distribution of rational points on algebraic planar curves, Chapter 1 provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is to develop a theory of approximation in the so far unstudied case when such curves are not defined by integer polynomials. A new concept of so–called “liminf sets” is then introduced in Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities. In short, in this type of problem, one prescribes the set of integers which the denominators of all the possible rational approximants of a given vector have to belong to. Finally, a reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This provides the first example of a Khintchine type result in the context of so–called uniform approximation