Linear-fractional invariance of multidimensional continued fractions (Q1992029)

The invariance of the simplex-karyon algorithm for expanding real numbers \(\alpha=(\alpha_1,\alpha_2,\dots,\alpha_d)\) in multidimensional continued fractions under linear-fractional transformations \(\alpha'=(\alpha_1',\alpha_2',\dots,\alpha_d')=U\langle \alpha\rangle\) with matrices \(U\) from the unimodular group \(\mathrm{GL}_{d+1}(\mathbb{Z})\) is established. It is proved that there exists a positive integer \(n_{\alpha,U,\eta^*}\) for which \[ \left| \alpha'_1-\frac{P_{n1}'}{Q_{n1}'} \right|+\dots +\left| \alpha'_d-\frac{P_{nd}'}{Q_{nd}'} \right|\leq \frac{c'}{|Q_{n}'|^{1+\eta^*}} , \quad n \geq n_{\alpha,U,\eta^*}. \] The approximations are the best ones for the new point \(\alpha'\) but with respect the \(s'_n\)-norms, for which the simplices \(s'_n=U \langle\alpha \rangle\), also possessing the contraction property and the minimality property, are chosen as the convex bodies.