The Hausdorff Dimension of Sets of Points Whose Simultaneous Rational Approximation by Sequences of Integer Vectors Have Errors with Small Product

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Abstract

Let m, n, be positive integers and let Q be an infinite subset of Zn. For any number t, define the set [formula]. It is shown in this paper that if ? ? [0, n] is the unique number such that the series [formula] is convergent when e{lunate} > 0 but divergent when e{lunate} < 0, then the Hausdorff dimension of the set EQ(m, n; t) is dim EQ(m, n; t) = mn - 1 + (1 + ?)/(1 + t), for all t > ?. © 1994 Academic Press. All rights reserved.

Original languageEnglish
Pages (from-to)75-79
Number of pages5
JournalJournal of Number Theory
Volume48
Issue number1
DOIs
Publication statusPublished - Jul 1994

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Simultaneous Approximation
Rational Approximation
Hausdorff Dimension
Set of points
Integer
Subset
Series

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abstract = "Let m, n, be positive integers and let Q be an infinite subset of Zn. For any number t, define the set [formula]. It is shown in this paper that if ? ? [0, n] is the unique number such that the series [formula] is convergent when e{lunate} > 0 but divergent when e{lunate} < 0, then the Hausdorff dimension of the set EQ(m, n; t) is dim EQ(m, n; t) = mn - 1 + (1 + ?)/(1 + t), for all t > ?. {\circledC} 1994 Academic Press. All rights reserved.",
author = "Rynne, {B. P.}",
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N2 - Let m, n, be positive integers and let Q be an infinite subset of Zn. For any number t, define the set [formula]. It is shown in this paper that if ? ? [0, n] is the unique number such that the series [formula] is convergent when e{lunate} > 0 but divergent when e{lunate} < 0, then the Hausdorff dimension of the set EQ(m, n; t) is dim EQ(m, n; t) = mn - 1 + (1 + ?)/(1 + t), for all t > ?. © 1994 Academic Press. All rights reserved.

AB - Let m, n, be positive integers and let Q be an infinite subset of Zn. For any number t, define the set [formula]. It is shown in this paper that if ? ? [0, n] is the unique number such that the series [formula] is convergent when e{lunate} > 0 but divergent when e{lunate} < 0, then the Hausdorff dimension of the set EQ(m, n; t) is dim EQ(m, n; t) = mn - 1 + (1 + ?)/(1 + t), for all t > ?. © 1994 Academic Press. All rights reserved.

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