On the $t$-adic Littlewood Conjecture

Abstract

The $p$-adic Littlewood Conjecture due to De Mathan and Teuli\'e asserts that for any prime number $p$ and any real number $\alpha$, the equation $$\inf_{|m|\ge 1} |m|\cdot |m|_p\cdot |\langle m\alpha \rangle|\, =\, 0 $$ holds. Here, $|m|$ is the usual absolute value of the integer $m$, $|m|_p$ its $p$-adic absolute value and $ |\langle x\rangle|$ 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 $\mathbb{K}\left(\left(t^{-1}\right)\right)$ of a ground field $\mathbb{K}$. This is the so-called \emph{$t$-adic Littlewood Conjecture} ($t$-LC). It is known that $t$--LC fails when the ground field $\mathbb{K}$ is infinite. This article is concerned with the much more difficult case when the latter field is finite. More precisely, a \emph{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 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 $\mathbb{F}_3$, the so-called Number Wall of this sequence, can be obtained as a two-dimensional automatic tiling satisfying a finite number of suitable local constraints.