2018/06/12 by Adiceam, Faustin, Nesharim, Erez, Lunnon, Fred · 2 citations
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1806.04478
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 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 \mathbbK((t-1)) of a ground field \mathbbK. This is the so-called t-adic Littlewood Conjecture (t-LC). It is known that t--LC fails when the ground field \mathbbK 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 \mathbbK 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 \mathbbF3, 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.