The distribution of close conjugate algebraic numbers

We investigate the distribution of real algebraic numbers of a fixed degree having a close conjugate number, the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds of Bugeaud and Mignotte. So far the results a la Bugeaud and Mignotte relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. The applications of our main theorem considered in this paper include generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.Comment: 17 pages, corrected versio

A Fr\'{e}chet law and an Erd\"os-Philipp law for maximal cuspidal windings

In this paper we establish a Fr\'{e}chet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying Extreme Value Theory. Subsequently, we show that this law gives rise to an Erd\"os-Philipp law and to various generalised Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erd\"os

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

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.Comment: 19 page

Integral polynomials with small discriminants and resultants

Let n∈N be fixed, Q>1 be a real parameter and Pn(Q) denote the set of polynomials over Z of degree n and height at most Q. In this paper we investigate the following counting problems regarding polynomials with small discriminant D(P) and pairs of polynomials with small resultant R(P 1, P 2):(i)given 0≤v≤n-1 and a sufficiently large Q, estimate the number of polynomials P∈Pn(Q) such that0<|D(P)|≤Q 2n-2-2v; (ii)given 0≤w≤n and a sufficiently large Q, estimate the number of pairs of polynomials P1,P2∈Pn(Q) such that0<|R(P1,P2)|≤Q 2n-2w. Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R. C. Vaughan and S. Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials

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

Diophantine Approximation and applications in Interference Alignment

This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of R n. The importance of such quantitative statements is explicitly discussed in Jafar's monograph [12, §4.7.1]

Badly approximable points on manifolds

Addressing a problem of Davenport we show that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate manifold in has full dimension. This also extends Schmidt's conjecture on badly approximable points to arbitrary dimensions