Sanju Velani

The ergodic theory of shrinking targets

SummaryTo any dynamical system equipped with a metric, we associate a class of “well approximable” sets. In the case of an expanding rational map of the Riemann sphere acting on its Julia set, we estimate and in some cases compute the Hausdorff dimension of the associated “well approximable” sets. The methods used show a clear link between distortion properties and the type of results obtained in this paper, via ergodic theory and ubiquity.

Diophantine approximation on planar curves: the convergence theory

The convergence theory for the set of simultaneously ψ-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in [1] and thereby completes the general metric theory for planar curves.

On a problem in simultaneous Diophantine approximation: Littlewood's conjecture

A consequence of Hurwitzs theorem is that the right-hand side of the above inequality cannot be improved by an arbitrary positive constant s. More precisely, for s < l / x / 5 there exist real numbers a E I for which the inequality Ilqall ~<cq -1 has at most a finite number of solutions. These a are the badly approximable numbers, and we will denote by B a d the set of all such numbers; that is,

On Simultaneously Badly Approximable Numbers

For any pair

An inhomogeneous transference principle and Diophantine approximation

i,j\ge 0

On shrinking targets for Zm actions on tori

with

A note on zero-one laws in metrical Diophantine approximation

i+j=1

Diophantine approximation and Hausdorff dimension in Fuchsian groups

let

A zero-infinity law for well-approximable points in Julia sets

{\mathbf Bad}(i,j)

Glasner sets and polynomials in primes

denote the set of pairs