On a problem in simultaneous Diophantine approximation: Littlewood's conjecture

Andrew Pollington; Sanju Velani

doi:10.1007/BF02392812

On a problem in simultaneous Diophantine approximation: Littlewood's conjecture

Andrew Pollington, Sanju Velani

Mathematics

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

Abstract

A well known conjecture of Littlewood in number theory states that $ q to infty q ||q| ||q| = 0 $ for any pair of real numbers $( $. It is easily seen, via the theory of continued fractions that the conjecture is true if either $ or $ are not in $ -- the set of badly approximable numbers. In view of this, the following natural question arises. Given $a in , are there any independent $b in for which the conjecture is true ? In attempting to answer this question we prove the following positive result. Given $a in , there exists a subset $ of $ with $dim G = 1$ such that for any $b in the inequality $ q ||q| ||q| leq 1 / log q $ is satisfied for infinitely many $q in . For such pairs $( in Bad $, this obviously implies the conjecture.

Original language	English
Pages (from-to)	287-306
Number of pages	20
Journal	Acta Mathematica
Volume	185
Issue number	2
DOIs	https://doi.org/10.1007/BF02392812
Publication status	Published - Sept 2001

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10.1007/BF02392812

Cite this

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title = "On a problem in simultaneous Diophantine approximation: Littlewood's conjecture",

abstract = "A well known conjecture of Littlewood in number theory states that $ q to infty q ||q| ||q| = 0 $ for any pair of real numbers $( $. It is easily seen, via the theory of continued fractions that the conjecture is true if either $ or $ are not in $ -- the set of badly approximable numbers. In view of this, the following natural question arises. Given $a in , are there any independent $b in for which the conjecture is true ? In attempting to answer this question we prove the following positive result. Given $a in , there exists a subset $ of $ with $dim G = 1$ such that for any $b in the inequality $ q ||q| ||q| leq 1 / log q $ is satisfied for infinitely many $q in . For such pairs $( in Bad $, this obviously implies the conjecture.",

author = "Andrew Pollington and Sanju Velani",

year = "2001",

month = sep,

doi = "10.1007/BF02392812",

language = "English",

volume = "185",

pages = "287--306",

journal = "Acta Mathematica",

issn = "1871-2509",

publisher = "International Press of Boston, Inc.",

number = "2",

}

TY - JOUR

T1 - On a problem in simultaneous Diophantine approximation

T2 - Littlewood's conjecture

AU - Pollington, Andrew

AU - Velani, Sanju

PY - 2001/9

Y1 - 2001/9

N2 - A well known conjecture of Littlewood in number theory states that $ q to infty q ||q| ||q| = 0 $ for any pair of real numbers $( $. It is easily seen, via the theory of continued fractions that the conjecture is true if either $ or $ are not in $ -- the set of badly approximable numbers. In view of this, the following natural question arises. Given $a in , are there any independent $b in for which the conjecture is true ? In attempting to answer this question we prove the following positive result. Given $a in , there exists a subset $ of $ with $dim G = 1$ such that for any $b in the inequality $ q ||q| ||q| leq 1 / log q $ is satisfied for infinitely many $q in . For such pairs $( in Bad $, this obviously implies the conjecture.

AB - A well known conjecture of Littlewood in number theory states that $ q to infty q ||q| ||q| = 0 $ for any pair of real numbers $( $. It is easily seen, via the theory of continued fractions that the conjecture is true if either $ or $ are not in $ -- the set of badly approximable numbers. In view of this, the following natural question arises. Given $a in , are there any independent $b in for which the conjecture is true ? In attempting to answer this question we prove the following positive result. Given $a in , there exists a subset $ of $ with $dim G = 1$ such that for any $b in the inequality $ q ||q| ||q| leq 1 / log q $ is satisfied for infinitely many $q in . For such pairs $( in Bad $, this obviously implies the conjecture.

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DO - 10.1007/BF02392812

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JO - Acta Mathematica

JF - Acta Mathematica

IS - 2

ER -