On the t-adic Littlewood Conjecture
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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 _|m|> 1 |m|· |m|_p· |〈 mα〉| = 0 holds. Here, |m| is the usual absolute value of the integer m, |m|_p 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. This article is concerned with the much more difficult case when the latter 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 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 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.
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