On simultaneously badly approximable numbers

Andrew Pollington; Sanju Velani

doi:10.1112/S0024610702003265

On simultaneously badly approximable numbers

Andrew Pollington, Sanju Velani

Mathematics

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

Abstract

For any pair i,j= 0 with i+j=1 let Bad(i,j) denote the set of pairs (a,ß) ¿ R2 for which max{¿qa¿1/i¿qß|1/j}>c/q for all q ¿ N. Here c=c(a,ß) is a positive constant. If i=0 the set Bad(0, 1) is identified with R×Bad where Bad is the set of badly approximable numbers. That is, Bad(0, 1) consists of pairs (a, ß) with a ¿ R and ß ¿ Bad If j=0 the roles of a and ß are reversed. It is proved that the set Bad(1,0)nBad (0,1)n Bad(i,j) has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

Original language	English
Pages (from-to)	29-40
Number of pages	12
Journal	Journal London Math Soc
Volume	66
Issue number	1
DOIs	https://doi.org/10.1112/S0024610702003265
Publication status	Published - Aug 2002

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10.1112/S0024610702003265

Cite this

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title = "On simultaneously badly approximable numbers",

abstract = "For any pair i,j= 0 with i+j=1 let Bad(i,j) denote the set of pairs (a,{\ss}) ¿ R2 for which max{¿qa¿1/i¿q{\ss}|1/j}>c/q for all q ¿ N. Here c=c(a,{\ss}) is a positive constant. If i=0 the set Bad(0, 1) is identified with R×Bad where Bad is the set of badly approximable numbers. That is, Bad(0, 1) consists of pairs (a, {\ss}) with a ¿ R and {\ss} ¿ Bad If j=0 the roles of a and {\ss} are reversed. It is proved that the set Bad(1,0)nBad (0,1)n Bad(i,j) has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions. ",

author = "Andrew Pollington and Sanju Velani",

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T1 - On simultaneously badly approximable numbers

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AU - Velani, Sanju

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N2 - For any pair i,j= 0 with i+j=1 let Bad(i,j) denote the set of pairs (a,ß) ¿ R2 for which max{¿qa¿1/i¿qß|1/j}>c/q for all q ¿ N. Here c=c(a,ß) is a positive constant. If i=0 the set Bad(0, 1) is identified with R×Bad where Bad is the set of badly approximable numbers. That is, Bad(0, 1) consists of pairs (a, ß) with a ¿ R and ß ¿ Bad If j=0 the roles of a and ß are reversed. It is proved that the set Bad(1,0)nBad (0,1)n Bad(i,j) has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

AB - For any pair i,j= 0 with i+j=1 let Bad(i,j) denote the set of pairs (a,ß) ¿ R2 for which max{¿qa¿1/i¿qß|1/j}>c/q for all q ¿ N. Here c=c(a,ß) is a positive constant. If i=0 the set Bad(0, 1) is identified with R×Bad where Bad is the set of badly approximable numbers. That is, Bad(0, 1) consists of pairs (a, ß) with a ¿ R and ß ¿ Bad If j=0 the roles of a and ß are reversed. It is proved that the set Bad(1,0)nBad (0,1)n Bad(i,j) has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

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DO - 10.1112/S0024610702003265

M3 - Article

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SP - 29

EP - 40

JO - Journal London Math Soc

JF - Journal London Math Soc

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