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
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 language | English |
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Place of Publication | Durham, NC |
Publisher | Duke University Press |
Pages | 2371–2419 |
Number of pages | 48 |
Publication status | Published - 15 Jul 2021 |