Diophantine approximation and badly approximable sets. (English) Zbl 1098.11039
A real number \(x\) is called badly approximable if there is a positive constant \(c(x)\) such that \(| x-p/q| \geq c(x)/q^2\) for every rational \(p/q\). An old result of I. Jarník states that the Hausdorff dimension of badly approximable numbers equals 1. The authors consider a far-reaching generalization, where the points \(x\) belong to a compact subspace \(\Omega\) of a metric space \(X\) containing the support of a non-atomic finite measure, and the rational points \(p/q\) are replaced by a family of subsets of \(X\) called resonant sets. In this general setting the authors prove that under various natural conditions, the badly approximable subsets of \(\Omega\) have full Hausdorff dimension. There are applications from number theory (classical, complex, \(p\)-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).
Reviewer: Veikko Ennola (Turku)
MSC:
| 11J83 | Metric theory |
| 11J61 | Approximation in non-Archimedean valuations |
| 28A80 | Fractals |
| 37C45 | Dimension theory of smooth dynamical systems |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
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