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Journal of the London Mathematical Society 2006 73(2):355-366; doi:10.1112/S0024610706022617
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© The London Mathematical Society

On the Littlewood Conjecture in Simultaneous Diophantine Approximation

Boris Adamczewski and Yann Bugeaud

CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1 Bât. Braconnier, 21 avenue Claude Bernard, 69622 Villeurbanne cedex, France boris.adamczewski{at}math.univ-lyon1.fr
Université Louis Pasteur UFR de mathématiques, 7, rue René Descartes, 67084 Strasbourg cedex, France bugeaud{at}math.u-strasbg.fr

Received 15 March 2005. Revision received 20 July 2005.

For any given real number {alpha} with bounded partial quotients, we construct explicitly continuum many real numbers ß with bounded partial quotients for which the pair ({alpha}, ß) satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.


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