On the t-adic Littlewood Conjecture

Faustin Adiceam, Erez Nesharim, Fred Lunnon

    Research output: Preprint/Working paperWorking paper

    Abstract

    The p–adic Littlewood Conjecture due to De Mathan and Teulié asserts that for any prime number p and any real number α, the equation infm|≥1|m| · |m|p · |hmαi| = 0 holds. Here, |m| is the usual absolute value of the integer m, |m|p its p–adic absolute value and |hxi| denotes the distance from a real number x to the set of integers. This still open conjecture stands as a variant of the well–known Littlewood Conjecture. In the same way as
    the latter, it admits a natural counterpart over the field of formal Laurent series Kt−1of a ground field K. This is the so–called tadic Littlewood Conjecture (t–LC).
    It is known that t–LC fails when the ground field K is infinite. This article is concerned
    with the much more difficult case when the latter field is finite. More precisely, a fully explicit
    counterexample is provided to show that t–LC does not hold in the case that K is a finite field with characteristic 3. Generalizations to fields with characteristics different from 3 are
    also discussed.
    The proof is computer assisted. It reduces to showing that an infinite matrix encoding Hankel determinants of the Paper–Folding sequence over F3, the so–called Number Wall of this
    sequence, can be obtained as a two–dimensional automatic tiling satisfying a finite number of suitable local constrain
    Original languageEnglish
    Place of PublicationDurham, NC
    PublisherDuke University Press
    Pages2371–2419
    Number of pages48
    Publication statusPublished - 15 Jul 2021

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