On the t-adic Littlewood conjecture

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

$$\underset{|m|\ge 1}{inf}|m|\cdot |m{|}_p\cdot |⟨m\mathit{\alpha }⟩|=0$$

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 $\mathbb{K}((t^{-1}))$ 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.