Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich–Dickinson–Velani [6] and Vaughan–Velani [22]. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich–Zorin [5] in the divergence case.     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

An inhomogeneous wave equation and non-linear Diophantine approximation

A non-linear Diophantine condition involving perfect squares and arising from an inhomogeneous wave equation on the torus guarantees the existence of a smooth solution. The exceptional set associated with the failure of the Diophantine condition and hence of the existence of a smooth solution is studied. Both the Lebesgue and Hausdorff measures of this set are obtained.     

Modified Schmidt games and Diophantine approximation with weights

We show that the sets of weighted badly approximable vectors in Rn are winning sets of certain games, which are modifications of (α,β)-games introduced by W.M. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.     

Metric Diophantine approximation over a local field of positive characteristic

The conjectures of Sprind?uk 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.     

Diophantine approximation on rational quadrics

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.     

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Let K denote the middle third Cantor set and . Given a real, positive function ψ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given. Dedicated to Maurice Dodson on his retirement—finally!     

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).     

Metric Diophantine approximation and ‘absolutely friendly’ measures

Let W(ψ) denote the set of ψ-well approximable points in and let K be a compact subset of 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 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      

Diophantine approximation and Cantor sets

We provide an explicit construction of elements of the middle third Cantor set with any prescribed irrationality exponent. This answers a question posed by Kurt Mahler.     

Invariance Principles in Metric Diophantine Approximation

In [7], LeVeque proved a central limit theorem for the number of solutions p,q of

On fractal measures and diophantine approximation

We study diophantine properties of a typical point with respect to measures on Namely, we identify geometric conditions on a measure on guaranteeing that -almost every is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called friendly. Examples include smooth measures on nondegenerate manifolds; thus this paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.     

Palindromic Prefixes and Diophantine Approximation

This article is devoted to simultaneous approximation to ξ and ξ² by rational numbers with the same denominator, where ξ is an irrational non-quadratic real number. We focus on an exponent β₀(ξ) that measures the regularity of the sequence of all exceptionally precise such approximants. We prove that β₀(ξ) takes the same set of values as a combinatorial quantity that measures the abundance of palindromic prefixes in an infinite word w. This allows us to give a precise exposition of Roy’s palindromic prefix method. The main tools we use are Davenport-Schmidt’s sequence of minimal points and Roy’s bracket operation.     

Diophantine approximation with mild divisibility constraints

Given an irrational number α and a sequence B of coprime positive integers with the sum of inverses convergent, we investigate the problem of finding small values of nα, with n B-free.     

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. Received: 24 October 2000; in final form: 10 November 2001 / Published online: 17 June 2002     

On Real Quadratic Number Fields and Simultaneous Diophantine Approximation

 Here we provide a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to insure that they are elements of the same quadratic number field over ℚ. Such a condition has implications to simultaneous diophantine approximation. In particular, we deduce an improvement to Dirichlet’s Theorem in this context which, as an immediate consequence, shows the Littlewood Conjecture to hold for all numbers α and β both from . Specifically, for all such pairs we have . (Received 10 August 1998; in revised form 23 November 1998)     

Integer parts of powers of rational numbers

We prove that the sequence [ξ(5/4)ⁿ], n=1,2, . . . , where ξ is an arbitrary positive number, contains infinitely many composite numbers. A corresponding result for the sequences [(3/2)ⁿ] and [(4/3)ⁿ],n=1,2, . . . , was obtained by Forman and Shapiro in 1967. Furthermore, it is shown that there are infinitely many positive integers n such that ([ξ(5/4)ⁿ],6006)>1, where 6006=2·3·7·11·13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to ξ(5/3)ⁿ and to ξ(7/5)ⁿ, n∈ℕ. The corresponding sets of possible divisors are also described.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.