## Geodesic multidimensional continued fractions.(English)Zbl 0813.11040

Given any real vector $$\underline \vartheta = (\vartheta_ 1, \dots, \vartheta_ d) \in \mathbb{R}^ d$$, a $$d$$-dimensional continued fraction expansion is constructed as follows: One considers the continuously parametrized family of lattices $$\mathbb{Z}^{d+1} B_ t$$ with bases $$B_ t = B_ t (\underline \vartheta) = \left( \begin{smallmatrix} I & o \\ - \underline \vartheta & t \end{smallmatrix} \right)$$ where $$I$$ is the $$d \times d$$-identity matrix and $$o$$ is a zero-column. The family $$B_ t$$ forms a left coset of a one-parameter subgroup of $$\text{GL} (d+1, \mathbb{R})^ +$$ which makes it a geodesic with respect to a group-invariant affine connection on $$\text{GL} (d + 1, \mathbb{R})$$. Now Minkowski reduction, or rather a technical refinement of this, is applied to obtain (“lexicographically”) reduced bases $$\widetilde B_ t = P_ tB_ t$$ $$(P_ t$$ some integer transformation matrix).
The crucial phenomenon that makes the concept work is the existence of a well-defined sequence of critical values $$t_ 1 > t_ 2 > \cdots > t_ i > \cdots > 0$$ with the property that $$P_ t$$ is constant on each of the open intervals $$(t_{i+1}, t_ i)$$ and changes at their boundary points. This allows the introduction of an associated sequence of partial quotient matrices $$A^{(i)}$$ by putting $$P^{(i+1)} = A^{(i)} P^{(i)}$$, where the “convergent” $$P^{(i)} = (p_{k ,j}^{(i)})$$ $$(k,j = 1, \dots, d + 1)$$ is the matrix $$P_ t$$ associated with $$(t_{i+1}, t_ i)$$. It is proved that the “denominators” $$q_ k^{(i)}=p^{(i)}_{k,d+1}$$ are unique if $$P^{(i)} B_ t (\underline \vartheta)$$ is lexicographically reduced on $$(t_{i+1}, t_ i)$$. Further, the expansion terminates (there are only finitely many critical values $$t_ i)$$ if and only if all components of $$\underline \vartheta$$ are rational. Moreover, in each dimension $$d$$, there is only a finite set of possible partial quotient matrices for all $$\underline \vartheta \in \mathbb{R}^ d$$.
As a consequence of the properties of Minkowski reduction it is shown that for all $$\underline \vartheta \in \mathbb{R}^ d$$ the expansion is strongly convergent in the following sense: Let $$\underline w_ k^{(i)} (\underline \vartheta)$$ be the simultaneous approximation vector with (rational) components $$p_{k, 1}^{(i)} /q_ k^{(i)}, \dots, p_{k,d}^{(i)}/q_ k^{(i)}$$; then $$\underline w_ k^{(i)} (\underline \vartheta) \to \underline 0$$ as $$i \to \infty$$ and $$q_ k^{(i)} \| \underline w_ k^{(i)} (\underline \vartheta) - \underline \vartheta \| \to 0$$ as $$i \to \infty$$ for $$1 \leq k \leq r$$ (Here $$r$$ is the rank of $$(\vartheta_ 1, \dots, \vartheta_ d)$$ over $$\mathbb{Q}$$ and $$\| \cdot \|$$ means the Euclidean norm). The expansion has important diophantine approximation properties; in particular, it finds, for any $$\underline \vartheta$$ and any integer $$Q \geq 1$$, a best approximation (with respect to $$\| \cdot \|)$$ $$\underline w = (p_ 1/q, \dots, p_ d/q)$$ to $$\underline \vartheta$$ with $$1 \leq q \leq (d+1)^{1/2}Q$$ and $$q \| \underline w - \underline \vartheta \| \leq (d+1)^{1/2} Q^{-1/d}$$. The method generalizes to give continued fraction expansions finding good approximations to an arbitrary set of linear forms.
Reviewer: G.Ramharter (Wien)

### MSC:

 11J70 Continued fractions and generalizations
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