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 inf|m|≥1|m|⋅|m|p⋅|⟨mα⟩|=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 K((t−1)) of a ground field K. This is the so-called t-adic Littlewood conjecture (t-LC). It is known that t-LC fails when the ground field 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 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 F3, 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.