Best simultaneous Diophantine approximations of some cubic algebraic numbers
Let \(N\) be a norm on \(\mathbb R^2\) and let \(\theta\in\mathbb R^2\). The set \({\mathcal M}(\theta)\) of best approximations of \(\theta\) with respect to \(N\) is the set of vectors \(q\theta-P\), where \(q\geq 1\) is an integer such that \[ N(q\theta - P) =\min\{N(q\theta - A)\; ;\; A\in\mathbb Z^{2}\}< \min\{N(k\theta - B)\; ;\; B\in\mathbb Z^{2}\} \] for \(1\leq k\leq q-1\). Let \(X^{3}+a_{2}X^{2}+a_{1}X-1\in\mathbb Z[X]\) be a polynomial with coefficients \(a_{1}>0\) and \(a_{2}>0\) having a unique real root \(\zeta\). Define \[ B=\begin{pmatrix} a_{1}&-a_{2}&-1\cr -1&0&0\cr 0&1&0\end{pmatrix} \] and \(\theta=(\zeta,\zeta^{2})\). The first result of the author is that there exists a norm \(N\) on \(\mathbb R^{2}\) and a finite number \(X_{1},\ldots,X_{m}\) of elements in \({\mathcal M}(\theta)\) such that \({\mathcal M}(\theta)\) is the union of a finite set and of the \(m\) sets \(\{B^{n} X_{i}\; ;\; n\geq 0\}\), \(1\leq i\leq m\). This refines a result of \textit{W. W.~Adams} [Pac. J. Math. 30, 1--14 (1969; Zbl 0182.37803); J. Number Theory 1, 179--194 (1969; Zbl 0172.06501)] and \textit{J. C.~Lagarias} [Proc. Queen's Number Theory Conf. 1979, Queen's Pap. Pure Appl. Math. 54, 453--474 (1980; Zbl 0453.10035)]. The particular case \(a_{1}=a_{2}=1\) had been considered by [\textit{N. Chekhova, P. Hubert} and \textit{A. Messaoudi}, J. Théor. Nombres Bordx. 13, No. 2, 371--394 (2001; Zbl 1038.37010)]. Further, let \(K\) be a cubic number field with exactly one real embedding. The second result of this paper is that there exist a unit \(\zeta\) in the ring of integers of \(K\), two positive integers \(a_{1}\) and \(a_{2}\) and an Euclidean norm on \(\mathbb R^{2}\) such that the set of best approximation vectors of \(\theta=(\zeta,\zeta^{2})\) is \(\{B^{n} \theta\; ;\; n\geq 0\}\), where \(B\) is defined as before.
scientific article; zbMATH DE number 2184590
scientific article