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15 July 2021 On the t-adic Littlewood conjecture
Faustin Adiceam, Erez Nesharim, Fred Lunnon
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Duke Math. J. 170(10): 2371-2419 (15 July 2021). DOI: 10.1215/00127094-2020-0077

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

inf|m|1|m||m|p|mα|=0

holds. Here |m| is the usual absolute value of the integer m, |m|p is its p-adic absolute value, and |x| 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 K((t1)) of a ground field K. This is the so-called t-adic Littlewood conjecture (t-LC).

It is known that t-LC fails when the ground field K is infinite. The present article is concerned with the much more difficult case when this 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 characteristic other than 3 are also discussed.

The proof is computer-assisted. It reduces to showing that an infinite matrix encoding Hankel determinants of the paperfolding sequence over F3, the so-called number wall of this sequence, can be obtained as a 2-dimensional automatic tiling satisfying a finite number of suitable local constraints.

Dedication

In honorem Christiani Figuli (Christian Potier).

Citation

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Faustin Adiceam. Erez Nesharim. Fred Lunnon. "On the t-adic Littlewood conjecture." Duke Math. J. 170 (10) 2371 - 2419, 15 July 2021. https://doi.org/10.1215/00127094-2020-0077

Information

Received: 19 June 2018; Revised: 28 February 2020; Published: 15 July 2021
First available in Project Euclid: 21 June 2021

Digital Object Identifier: 10.1215/00127094-2020-0077

Subjects:
Primary: 11J04
Secondary: 11B85, 11J61, 37A17

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 10 • 15 July 2021
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