Differentiation of induced toric tilings and multi-dimensional approximations of algebraic numbers

We consider the induced tilings $\mathcal{T=T}|_\mathrm{Kr}$ of the $D$-dimensional torus $\mathbb T^D$ generated by embedded karyons $\mathrm{Kr}$. The differentiations $\sigma\colon\mathcal{T\to T}^\sigma$ are defined under which we obtaine again the induced tilings $\mathcal T^\sigma=\mathcal T|_{\mathrm{Kr}^\sigma}$ with a derivative karyon $\mathrm{Kr}^\sigma$. They are used for approximation of $0\in\mathbb T^D$ by an infinite sequence of points $x_j\equiv j\alpha\mod\mathbb Z^D$ for $j=0,1,2,\dots$, where $\alpha=(\alpha_1,\dots,\alpha_D)$ is vector whose coordinates $\alpha_1,\dots,\alpha_D$ belong to an algebraic field $\mathbb Q(\theta)$ of degree $D+1$ over the rational field $\mathbb Q$. For this purpose, we construct an infinite sequence of convex parallelohedra $T^{(i)}\subset\mathbb T^D$ for $i=0,1,2,\dots$ and define for them some natural oders $m^{(0)}<m^{(1)}<\dots<m^{(i)}<\dots$ Then the above parallelohedra contain a subsequence of points $\{x_{j'}\}_{j'=1}^\infty$ that give the best approximation of $0\in\mathbb T^D$.