Multiplicative \(p\)-adic approximation
The \(p\)-adic Littlewood conjecture of \textit{B. de Mathan} and \textit{O. Teulié} [Monatsh. Math. 143, No. 3, 229--245 (2004; Zbl 1162.11361)] states that for any real number \(\xi\), and every prime number \(p\), \[ \inf_{q \in \mathbb{Z}, q \ge 1} q \cdot \Vert q\xi \Vert \cdot \vert q \vert_p = 0, \] where \(\Vert \cdot \Vert\) denotes the distance to the nearest integer and \(\vert \cdot \vert_p\) denotes the \(p\)-adic absolute value. A related conjecture states that for every prime number \(p\) and every \(p\)-adic number \(x\), \[ \inf_{a,b \in \mathbb{Z}\setminus\{0\}} \vert ab\vert \cdot \vert ax-b\vert_p = 0. \] Both conjectures remain open, though it is known from the work of \textit{M. Einsiedler} and \textit{D. Kleinbock} [Compos. Math. 143, No. 3, 689--702 (2007; Zbl 1149.11036)] that the set of elements for which either conjecture fails must lie in a countable union of sets of box counting dimension \(0\). In the present paper, the authors are concerned with the second conjecture and the relation between the two. The first two results are metric in nature. Letting \(\log^*(n) = \max\{1, \log n\}\), it is shown that a strong form of the second conjecture holds for almost all \(x \in \mathbb{Q}_p\) with respect to the Haar measure, namely that for almost all \(x \in \mathbb{Q}_p\), \[ \inf_{a,b \in \mathbb{Z}\setminus\{0\}} \vert ab\vert \cdot \log^* \vert ab\vert \cdot \log^* \log^* \vert ab\vert \cdot \left\vert x-\frac{a}{b} \right\vert_p = 0. \] Subsequently, it is shown that this result fails for a set of maximal Hausdorff dimension. Secondly, a necessary and sufficient criterion for the second conjecture to hold for a \(p\)-adic number \(x \in \mathbb{Q}_p\) is derived in terms of the continued fractions of the truncated and suitably normalised Hensel expansions of \(x\). It is then shown that under the original \(p\)-adic Littlewood conjecture, this condition is satisfied, so that the first conjecture implies the second. Towards the end of the paper, the authors study the analogous conjectures over \(K((X^{-1}))\) with degree valuation, where \(K\) is some field. In this case, the analogue of the \(p\)-adic Littlewood conjecture is known to fail when \(K\) is infinite and when \(K = \mathbb{F}_3\). Here, it is shown that the analogue of the second conjecture fails when \(K\) is infinite; and that if the analogue of the \(p\)-adic Littlewood conjecture holds, then so does the analogue of the second conjecture. In the final section of the paper, the authors present some numerical evidence towards the second conjecture in the \(p\)-adic case, obtained by applying the necessary and sufficient criterion derived previously.
scientific article; zbMATH DE number 7488516
scientific article