Diophantine approximation in characteristiep

We study diophantine approximations to algebraic functions in characteristicp. We improve a theorem of Osgood, and give two classes of examples showing that this result is nearly sharp. One of these classes exhibits a new phenomenon.     

Diophantine approximation and self-conformal measures

It is proved that the Hausdorff measure on the limit set of a finite conformal iterated function system is strongly extremal, meaning that almost all points with respect to this measure are not multiplicatively very well approximable. This proves Conjecture 10.6 from (on fractal measures and Diophantine approximation, preprint, 2003). The strong extremality of all (S,P)-invariant measures is established, where S is a finite conformal iterated function system and P is a probability vector. Both above results are consequences of the much more general Theorem 1.5 concerning Gibbs states of Hölder families of functions.     

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.     

Diophantine approximation with multiplicative functions

Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that

Diophantine approximation with multiplicative functions

Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that $$\left|\frac{\sigma(n)}{n}-\alpha \right| < n^{-0.52}.$$ We prove this result by modifying an argument given by Wolke (Monatsh Math 83:163–166, 1977) which in its original form could not produce an exponent greater than 0.5. We also explain how the exponent can be improved to 0.61 on the Riemann Hypothesis.     

Nevanlinna theory and Diophantine approximation

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

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.     

Diophantine approximation with sign constraints

Let $\alpha $ and $\beta $ be real numbers such that $1$ , $\alpha $ and $\beta $ are linearly independent over $\mathbb {Q}$ . A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max \{|x_1|,|x_2|\}^{-2}$ . In 1976, Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma \cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb {Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max \{|x_1|,|x_2|\}^\gamma $ is arbitrarily small. Although Schmidt later conjectured that $\gamma $ can be replaced by any number smaller than $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,\alpha ,\beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper.     

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).     

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 exponents for mildly restricted approximation

We are studying the Diophantine exponent μ _n,l defined for integers 1≤l<n and a vector α∈ℝ ⁿ by letting

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     

Inhomogeneous Diophantine approximation on planar curves

In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367–426, 2007) and Vaughan and Velani (Invent Math 166:103–124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of ‘shifted’ rational points near planar curves.     

Modular forms and effective Diophantine approximation

After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey?s height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey?s height conjecture which applies to all elliptic curves over $\mathbb{Q}$, not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of ${\mathbb{P}}^1?\{0,1,\infty \}$), with a completely explicit and effective bound, without using any variant of Baker?s theory or the Thue–Bombieri method.