On a problem of K. Mahler: Diophantine approximation and Cantor sets

Jason Levesley; Cem Salp; Sanju L. Velani

doi:10.1007/s00208-006-0069-8

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Jason Levesley, Cem Salp, Sanju L. Velani

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let K denote the middle third Cantor set and . Given a real, positive function ¿ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x - p/q| < ¿(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . 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. Mathematics Subject Classification (2000) Primary 11J83 - Secondary 11J82 - Secondary 11K55 Dedicated to Maurice Dodson on his retirement—finally!

Original language	English
Pages (from-to)	97-118
Number of pages	21
Journal	Mathematische Annalen
Volume	338
Issue number	1
DOIs	https://doi.org/10.1007/s00208-006-0069-8
Publication status	Published - May 2007

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10.1007/s00208-006-0069-8

http://dx.doi.org/10.1007/s00208-006-0069-8

Cite this

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title = "On a problem of K. Mahler: Diophantine approximation and Cantor sets",

abstract = "Let K denote the middle third Cantor set and . Given a real, positive function ¿ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x - p/q| < ¿(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . 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{\textquoteright}s assertion are also given. Mathematics Subject Classification (2000) Primary 11J83 - Secondary 11J82 - Secondary 11K55 Dedicated to Maurice Dodson on his retirement—finally!",

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journal = "Mathematische Annalen",

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T1 - On a problem of K. Mahler: Diophantine approximation and Cantor sets

AU - Levesley, Jason

AU - Salp, Cem

AU - Velani, Sanju L.

PY - 2007/5

Y1 - 2007/5

N2 - Let K denote the middle third Cantor set and . Given a real, positive function ¿ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x - p/q| < ¿(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . 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. Mathematics Subject Classification (2000) Primary 11J83 - Secondary 11J82 - Secondary 11K55 Dedicated to Maurice Dodson on his retirement—finally!

AB - Let K denote the middle third Cantor set and . Given a real, positive function ¿ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x - p/q| < ¿(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . 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. Mathematics Subject Classification (2000) Primary 11J83 - Secondary 11J82 - Secondary 11K55 Dedicated to Maurice Dodson on his retirement—finally!

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DO - 10.1007/s00208-006-0069-8

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JF - Mathematische Annalen

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