On the t-adic Littlewood Conjecture
On the t-adic Littlewood Conjecture
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DOI：
10.1215/00127094-2020-0077
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发表时间：
2018-06
期刊：
影响因子：
--
通讯作者：
F. Adiceam;Erez Nesharim;Fred Lunnon
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文献类型：
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作者：
F. Adiceam;Erez Nesharim;Fred Lunnon
由De Mathan和Teulie提出的$p$-adic Littlewood猜想断言，对于任何素数$p$和任何真实的数$\alpha$，方程$\inf_{|M| {\fnarial black\fs12\bord1\shad0\4aH00\fscx90\fscy110}| M|\c点|M|_p\cdot|\langle m\alpha \rangle|\，=\，0 $$保持不变。这里$|M| $是整数$m$的通常的绝对值，$|M|_p$它的$p$-adic绝对值和$|\langle x\rangle| $表示从一个真实的数$x$到整数集合的距离。这个仍然开放的猜想是著名的Littlewood猜想的一个变体。以与后者相同的方式，它允许一个自然对应物在一个基域$\mathbb {K}$的形式洛朗级数$\mathbb {K}\left（\left（t^{-1}\right）\right）$的域上。这就是所谓的\n {$t$-adic Littlewood猜想}（$t$-LC）。已知当地面场$\mathbb{K}$为无穷大时，$t$--LC失效。这篇文章关注的是更困难的情况下，后者领域是有限的。更确切地说，一个完全显式的反例表明，$t$-LC不成立的情况下，$\mathbb{K}$是一个有限域的特征3。还讨论了具有不同于3的特征的场的推广。证明是计算机辅助的。它减少到显示，一个无限矩阵编码的Hankel行列式的纸折叠序列在$\mathbb{F}_3$，这个序列的所谓的数字墙，可以得到一个二维自动平铺满足有限数量的适当的局部约束。
The $p$-adic Littlewood Conjecture due to De Mathan and Teulie asserts that for any prime number $p$ and any real number $\alpha$, the equation $$\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.