The geometry of badly approximable vectors
Let \({\mathbf v}= (v_1,v_2,\dots, v_k)\), where \(v_1,v_2,\dots, v_k\in \mathbb{R}\), be a \(k\)-dimensional vector. For vectors whose components are ``badly approximable'' by rationals in some sense, the distribution of \(nv\) \((n=1,2,\dots)\) modulo the \(k\)-dimensional unit-cube is investigated. NEWLINENEWLINENEWLINEUnfortunately, the paper contains inaccuracies which make its understanding impossible. To begin with the definition of ``badly approximability'', the author defines it by NEWLINE\[NEWLINE\|mv_j\|\geq c/m^{(1/k)} \quad\text{for }m=1,2,\text{ and for all }1,2,\dots, k.\tag \(*\) NEWLINE\]NEWLINE Such a \({\mathbf v}\) does not exist; namely, for all components \(v_j\) of \({\mathbf v}\) there is a natural \(m\) with \(\|mv_j\|< c/m\). (As usual, \(\|\cdot\|\) denotes the distance from the nearest integer.) Instead of \((*)\), the correct definition of the ``badly approximability'' should be: \({\mathbf v}= (v_1,v_2,\dots, v_k)\) is called badly approximable if for all natural \(m\) we have \(\max\|mv_j\|>c/m^{(1/k)}\), where the maximum has to be taken over all components of the vector \({\mathbf v}\). NEWLINENEWLINENEWLINEAnother inaccuracy, given in the introduction is, that if all components are irrational, the sequence \(m{\mathbf v}\) \((m=1,2,\dots)\) is uniformly distributed modulo the \(k\)-dimensional unit cube. A counter-example in the two-dimensional case: \(v_1= \root 2\of 2\), \(v_2= 2\root 2\of 2\). All points are on the two straight lines from \((0,0)\) to \((.5,1)\) and from \((.5,0)\) to \((1,1)\). The necessary and sufficient condition for the uniform distribution is the linear independence of 1 and the components of \({\mathbf v}\).
scientific article; zbMATH DE number 1679000