Geodesic multidimensional continued fractions. (English) Zbl 0813.11040

Lagarias, J. C.

Proc. Lond. Math. Soc., III. Ser. 69, No. 3, 464-488 (1994).

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

Keywords:

multidimensional continued fractions; simultaneous rational approximation; geodesic; Minkowski reduction