Theory Of Diophantine Approximation Research Articles

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C 3 non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C 1 (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.

An inhomogeneous transference principle and Diophantine approximation

In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research. However, the techniques developed to date do not seem to be applicable to inhomogeneous approximation. Consequently, the theory of inhomogeneous Diophantine approximation on manifolds remains essentially non-existent. In this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones. This is rather surprising as the inhomogeneous theory contains the homogeneous theory and so is more general. As a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Furthermore, we prove a complete inhomogeneous version of the profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of friendly measures. The results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory.

Logarithm laws and shrinking target properties

We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation and probability theory.

Large intersection properties in Diophantine approximation and dynamical systems

We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle and in the perturbation theory for Hamiltonian systems.

Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan)

Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

Metric Diophantine approximation over a local field of positive characteristic

The conjectures of Sprindžuk in the metric theory of Diophantine approximation are established over a local field of positive characteristic. In the real case, these were settled by D. Kleinbock and G.A. Margulis using a new technique which involved nondivergence estimates for quasi-polynomial flows on the space of lattices. We extend their technique to the positive characteristic setting.

Nevanlinna theory and diophantine approximation

We discuss the analogue of the Nevanlinna theory and the theory of Diophantine approximation, focussing on the second main theorem and abc-conjecture.

Metric Diophantine approximation and ‘absolutely friendly’ measures

Let W(ψ) denote the set of ψ-well approximable points in \(\mathbb{R}^{d} \) and let K be a compact subset of \(\mathbb{R}^{d} \) which supports a measure μ. In this short article, we show that if μ is an ‘absolutely friendly’ measure and a certain μ-volume sum converges then \(\mu\,(W(\psi)\,\cap\,K)\,=\,0.\) The result obtained is in some sense analogous to the convergence part of Khintchine’s classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced in [2] and includes measures supported on self-similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of \(W(\psi)\,\cap\,K.\)

The Distribution of Geodesic Excursions Out the End of a Hyperbolic Orbifold and Approximation with Respect to a Fuchsian Group

A generic geodesic on a finite area, noncompact surface with a hyperbolic orbifold structure travels densely about the surface, making excursions out the non-compact end and then returning to a compact piece of the surface. We show that for almost all geodesics on such a surface there is a limiting distribution for the sequence of depths of the excursions out a noncompact end. The distribution is independent of the surface and closely resembles the distribution arising from a sequence of continued fraction approximations in the metrical theory of diophantine approximation. This metrical theory can then be reproduced in the setting of approximation by parabolic fixed points in a Fuchsian group.

Value distribution theory and Diophantine approximation

In this paper, we will introduce some problems and results between Diophantine approximation and value distribution theory.

Diophantine approximation and badly approximable sets

Let ( X , d ) be a metric space and ( Ω , d ) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω . Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad ( i , j ) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p -adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).

On a theorem of V. Bernik in the metric theory of Diophantine approximation

1. Introduction. We begin by introducing some notation: #S will denote the number of elements in a finite set S; the Lebesgue measure of a measurable set S ⊂ R will be denoted by |S|; Pn will be the set of integral polynomials of degree 6 n. Given a polynomial P, H(P) will denote the height of P, i.e. the maximum of the absolute values of its coefficients;Pn(H) = {P ∈ Pn : H(P) = H}. The symbol of Vinogradov ≪ in the expression A ≪ B means A 6 CB, where C is a constant. The symbol ≍ means both ≪ and ≫. Given a point x ∈ R and a set S ⊂ R, dist(x,S) = inf{|x −s| : x ∈ S}. Throughout, � will be a positive function. Mahler’s problem. In 1932 K. Mahler [10] introduced a classification of real numbers x into the so-called classes of A,S,T and U numbers according to the behavior of wn(x) defined as the supremum of w > 0 for which |P(x)| n for all x ∈ R. Mahler [9] proved that for almost all x ∈ R (in the sense of Lebesgue measure) wn(x) 6 4n, thus almost all x ∈ R are in the S-class. Mahler has also conjectured that for almost all x ∈ R one has the equality wn(x) = n. For about 30 years the progress in Mahler’s problem was limited to n = 2 and 3 and to partial results for n > 3. V. Sprindzuk proved Mahler’s conjecture in full (see [12]). A. Baker’s conjecture. Let Wn(�) be the set of x ∈ R such that there are infinitely many P ∈ Pn satisfying (1) |P(x)| < �(H(P)).

Open Diophantine Problems

Diophantine Analysis is a very active domain of mathemat- ical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai's Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel's Conjecture). Some questions related to Mahler's measure and Weil absolute logarithmic height are then considered (e.g., Lehmer's Problem). We also discuss Mazur's question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields. 2000 Math. Subj. Class. Primary 11Jxx; Secondary 11Dxx, 11Gxx, 14Gxx.

Systematic construction of full diversity algebraic constellations

A simple and systematic approach for constructing full diversity m-dimensional constellations, carved from lattices over a number ring R, is proposed for an arbitrary dimension m. When R=Z[wn], the nth cyclotomic number ring, all the possible dimensions that allow for achieving the optimal minimum product distances using the proposed approach are determined. It turns out that one can construct optimal unitary transformations using our construction if and only if m factors into a power of 2 and powers of the primes dividing n. For m not satisfying these conditions, a method based on Diophantine approximation theory is proposed to optimize the minimum product distance. A lower bound on the product distance is given in this case, thus ensuring full diversity with good minimum product distances. Furthermore, the proposed approach subsumes the optimal unitary transformations proposed by Giraud et al. over R=Z[w4] and R=Z[w3], while giving optimal unitary transformations for infinitely many new values of n and m.

Groupes g�om�triquement finis d?automorphismes d?arbres et approximation diophantienne dans les arbres

We introduce a class of non uniform tree lattices, analogous to the fundamental group of geometrically finite negatively curved manifolds. This class contains all the algebraic examples. Given such a tree lattice, we develop, as in [HP2] in the manifold case, a theory of diophantine approximation of points in the boundary of the tree by the parabolic fixed points. We compute approximation constants for some algebraic examples. All results are expressed intrinsically in terms of a natural geodesic flow for a graph of groups.

Universal space-time coding

A universal framework is developed for constructing full-rate and full-diversity coherent space-time codes for systems with arbitrary numbers of transmit and receive antennas. The proposed framework combines space-time layering concepts with algebraic component codes optimized for single-input-single-output (SISO) channels. Each component code is assigned to a "thread" in the space-time matrix, allowing it thus full access to the channel spatial diversity in the absence of the other threads. Diophantine approximation theory is then used in order to make the different threads "transparent" to each other. Within this framework, a special class of signals which uses algebraic number-theoretic constellations as component codes is thoroughly investigated. The lattice structure of the proposed number-theoretic codes along with their minimal delay allow for polynomial complexity maximum-likelihood (ML) decoding using algorithms from lattice theory. Combining the design framework with the Cayley transform allows to construct full diversity differential and noncoherent space-time codes. The proposed framework subsumes many of the existing codes in the literature, extends naturally to time-selective and frequency-selective channels, and allows for more flexibility in the tradeoff between power efficiency, bandwidth efficiency, and receiver complexity. Simulation results that demonstrate the significant gains offered by the proposed codes are presented in certain representative scenarios.

An analogue of a theorem of Szüsz for formal Laurent series over finite fields

About 40 years ago, Szüsz proved an extension of the well-known Gauss–Kuzmin theorem. This result played a crucial role in several subsequent papers (for instance, papers due to Szüsz, Philipp, and the author). In this note, we provide an analogue in the field of formal Laurent series and outline applications to the metric theory of continued fractions and to the metric theory of diophantine approximation.

Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation

We study the solutions in s of a “Dirichlet polynomial equation” miri + … + m M r s M = 1. We distinguish two cases. In the lattice case, when r j = r k j are powers of a common base r, the equation corresponds to a polynomial equation, which is readily solved numerically by using a computer. In the nonlattice case, when some ratio log r j / log r 1, j ≥ 2, is irrational, we obtain information by approximating the equation by lattice equations of higher and higher degree. We show that the set of lattice equations is dense in the set of all equations, and deduce that the roots of a nonlattice Dirichlet polynomial equation have a quasiperiodic structure, which we study in detail both theoretically and numerically. This question is connected with the study of the complex dimensions of self-similar strings. Our results suggest, in particular, that a nonlattice string possesses a set of complex dimensions with countably many real parts (fractal dimensions) which are dense in a connected interval. Moreover, we find dimension-free regions of nonlattice self-similar strings. We illustrate our theory with several examples. In the long term, this work is aimed in part at developing a Diophantine approximation theory of (higher-dimensional) selfsimilar fractals, both qualitatively and quantitatively.

Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C k manifold in R n , for any k,m,n∈ N , and for any τ>0 let S τ(M)={ x ∈M : ||q x ||<q −τ for infinitely many q∈ N}, where, for x∈ R , ||x|| min{|x−i| : i∈ Z}, and for x =(x 1,…,x n)∈ R n , || x || = max{||x 1||,…,||x n||}. In this paper it will be shown that for any C k manifold M there exist C k manifolds M z, M p arbitrarily ‘ C k -close’ to M with the property that, for all sufficiently large τ, dim S τ(M z )=0, dim S τ(M p )>0. This result shows that the non-zero curvature conditions which have been successfully used to tackle other aspects of the theory of Diophantine approximation on manifolds are unable to distinguish between these two cases when we look at simultaneous Diophantine approximation.

Diophantine approximation for negatively curved manifolds

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant in its moduli space.