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
holds. Here is the usual absolute value of the integer m, is its p-adic absolute value, and 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 of a ground field . This is the so-called t-adic Littlewood conjecture (t-LC).
It is known that t-LC fails when the ground field 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 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 , 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.
In honorem Christiani Figuli (Christian Potier).
"On the t-adic Littlewood conjecture." Duke Math. J. 170 (10) 2371 - 2419, 15 July 2021. https://doi.org/10.1215/00127094-2020-0077