The karyon algorithm for expansion in multidimensional continued fractions (Q2011296)

Let \(\alpha=(\alpha_1,\dots,\alpha_d)\) be points with arbitrary real coordinates. Let \(\Delta_e=\Delta_e^d\) be the closed \(d\)-dimensional unit simplex with vertices at the points \(e_0=(0,\dots,0),e_1=(1,0,\dots,0),\dots,\) \(e_d=(0,0,\dots,1)\) of the space \(\mathbb{R}^d. \) The scheme of a simplex-karyon algorithm is follows. An objective function \(\varrho\) is selected. This function and the point \(\alpha\) determine a certain \(\varrho\)-strategy for constructing an infinite monotone sequence \[ \mathbf{s} = \mathbf{s}_0 \supset \mathbf{s}_1 \supset \dots \supset \mathbf{s}_n \supset \dots \ni \alpha,\tag{1} \] which consists of open \(d\)-dimensional simplices \(s_n\) with rational vertices. A universal karyon algorithm, which applies to arbitrary real points \(\alpha\) and is a modification of the simplex-karyon algorithm, is suggested in the paper. The main distinction of the karyon algorithm is that instead of the simplex sequence (1), it considers an infinite sequence \( \mathbf{T}=\mathbf{T}_0 \to \mathbf{T}_1 \to \dots \to \mathbf{T}_n \to \dots \) of \(d\)-dimensional parallelohedra \(\mathbf{T}_n\), which are, in general, not interrelated by inclusion, as in the sequence (1). Every parallelohedron \(\mathbf{T}_n\) is obtained from the preceding one \(\mathbf{T}_{n-1}\) by differentiation, i.e., \[ \mathbf{T}_n=\mathbf{T}_{n-1}^{\sigma_n}.\tag{2} \] It is proved that the parallelohedra \(\mathbf{T}_n\) occurring in the sequence (2) also possess a minimality property. An estimate for the rate of approximation of real numbers \(\alpha\) by multidimensional continued fractions is derived.