The mixed Littlewood conjecture for pseudo-absolute values

Stephen Harrap, Alan Haynes

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In this paper we study the Mixed Littlewood Conjecture with pseudo-absolute values. We show that if p is a prime and D is a pseudo-absolute value sequence satisfying mild conditions then then the infimum over natural numbers n of the quantity n.|n|_p.|n|_D.||nx|| equals 0 for all real x. Our proof relies on a measure rigidity theorem due to Lindenstrauss and lower bounds for linear forms in logarithms due to Baker and Wustholz. We also deduce the answer to the related metric question of how fast the infimum above tends to zero, for almost every x.
Original languageEnglish
Pages (from-to)1-20
Number of pages20
JournalMathematischen Annalen
Issue numbern/a
Early online date26 Mar 2013
Publication statusPublished - Nov 2013

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