On a problem of K. Mahler: Diophantine approximation and Cantor sets
By Jason Levesley, Cem Salp, Sanju Velani. 2005
92 citations recorded by the source corpus.
Let K denote the middle third Cantor set and $$\mathcal{A}:= \{ 3^{n} : n = 0,1,2, \ldots \}$$. Given a real, positive function ψ let $$ W_{\mathcal{A}}(\psi)$$ denote the set of real numbers x in the unit interval for which there exist infinitely many $$(p,q) \in \mathbb{Z} \times {\mathcal{A}}$$ such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for $$W_{\mathcal{A}}(\psi) \cap K$$. 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.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Theoretical Computer Science
View Paper: https://www.semanticscholar.org/paper/a1ad75689899e0c091e2664c15ba9730464bdb9d
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In Mathematical Physics
In Statistical and Nonlinear Physics
In Theoretical Computer Science
Citations in the corpus
- Diophantine approximation and Cantor sets (2008, methodology): “7 of [11].”