Faustin Adiceam, Erez Nesharim, Fred Lunnon

Duke Math. J. Advance Publication, 1-49, (2021) DOI: 10.1215/00127094-2020-0077
KEYWORDS: Littlewood Conjecture, p-adic Littlewood conjecture, t-adic Littlewood conjecture, number walls, Hankel determinants, Paperfolding sequence, automatic sequences, automatic tilings, substitution tilings, linear complexity, finite fields, Padé approximation, Padé table, dragon sequence, pagoda sequence, 11J04, 11J61, 11B85, 37A17

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

$$\underset{|m|\ge 1}{inf}|m|\cdot |m{|}_{p}\cdot |\u27e8m\mathit{\alpha}\u27e9|=0$$

holds. Here $|m|$ is the usual absolute value of the integer *m*, $|m{|}_{p}$ is its *p*-adic absolute value, and $|\u27e8x\u27e9|$ 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 $\mathbb{K}(({t}^{-1}))$ of a ground field $\mathbb{K}$. This is the so-called *t-adic Littlewood conjecture* (*t*-LC).

It is known that *t*-LC fails when the ground field $\mathbb{K}$ 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 $\mathbb{K}$ 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 ${\mathbb{F}}_{3}$, 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.