Diophantine approximation and badly approximable sets

By S. Kristensen, R. Thorn, Sanju Velani. 2004

87 citations recorded by the source corpus.

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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7c9305afc22f29742a2c28a1aa2dba7e29b8d7a0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Badly approximable points on manifolds (2013, background): “It is worth mentioning that the setsBad(r) have been studied at length in all dimensions and for arbitrary collections of weights, see [22,34–36,41].”
Metric Diophantine Approximation : aspects of recent work (2016, background): “, in) have long been know to be uncountable and indeed of full dimension, see [41, 67, 68, 76], the following conjecture of Schmidt dating back to 1982 remained unresolved until reasonably recently.”
On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture (2010, background): “For further background and various strengthenings of this full dimension statement the reader is referred to [4, 5, 7].”
The set of badly approximable vectors is strongly C1 incompressible (2011, result): “Games played on proper subsets An application of Schmidt games to proving abundance of badly approximable vectors on certain fractal subsets of R first appeared in [12], and was motivated by earlier related results [33, 17, 18, 21].”
Diophantine approximation and Cantor sets (2008, methodology): “Furthermore, metric number theory was used in [7,8,10] to show that K contains badly approximable numbers, that is, numbers ξ for which |ξ − p/q| > c/q2 holds for every rational p/q and some positive constant c.”
On a problem of K. Mahler: Diophantine approximation and Cantor sets (2005, background): “It is well known that Bad ∩ K is non-empty and moreover that dim(Bad ∩ K) = γ—see for example [10, 11 ].”
Schmidt’s game on fractals (2009, methodology): “Answers to both of these questions have been independently given in [KW] and [ KTV ] proving that dimK\BA=dimK for a large family of sets including those mentioned above.”