Abstract
The problem of simultaneously approximating a vector of irrational numbers with rationals is analyzed in a geometrical setting using notions of dynamical systems theory. We discuss here a (vectorial) multidimensional continued-fraction algorithm (MCFA) of additive type, the generalized mediant algorithm (GMA), and give a geometrical interpretation to it. We calculate the invariant measure of the GMA shift as well as its Kolmogorov-Sinai (KS) entropy for arbitrary number of irrationals. The KS entropy is related to the growth rate of denominators of the Euclidean algorithm. This is the first analytical calculation of the growth rate of denominators for any MCFA.
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Abbreviations
- L + :
-
set of positive integers
- [.]:
-
Gauss integer symbol (Section 2)
- h :
-
entropy
- I :
-
of irrationals to be simultaneously approximated
- d :
-
dimension of the vector of convergents (equal to I+1)
- ℐ P :
-
unit hypercube inp dimensions
- ℒ :
-
support of the invariant measure (see Section 5)
- Eij :
-
elementary matrix, with klth componentδ kl +δ ik δ jl
- E-string:
-
product of elementary matrices given by the algorithm
- verticesV i :
-
corners of the elementary simplex adjoined to the origin (Section 3)
- mediantsM ik :
-
a direct sum of any two of the vertices (Section 3)
- focus:
-
sum of all the vertices (Section 3)
- Euclidean:
-
reverse of the E-string procedure (see Section 2) algorithm
- OCF:
-
ordinary continued-fraction algorithm
- GMA:
-
generalized mediant algorithm: the subject of this paper
- JP:
-
Jacobi-Perron: the most well-studied MCFA
- MCFA:
-
Multidimensional continued-fraction algorithm
- KS entropy:
-
Kolmogorov-Sinai entropy
- T OCF :
-
ordinary continued-fraction shift map
- FS :
-
Farey shift map
- (a,..., z) :
-
irrational vector withI components; each element is an irrational
- dμ(x) :
-
invariant measure
- ρ(x) :
-
invariant density =dμ(x)/dx]
- λ 1, λ2,...,λ d :
-
thed Oseledec eigenvalues of the E-string (see Section 4) ordered λ1>1>λ2⩾λ3⩾...
- σ1,...,σd−1 :
-
Oseledec eigenvalues of the shift map (Section 4) ordered greatest to smallest; all the σi>1, and σi=λ1/λd i+1
- ln σ1,..., In σd− 1 :
-
Oseledecexponents of the shift map (Section 4)
- Perm:
-
a permutation matrix (Section 4)
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P. R. Baldwin, A convergence exponent for multidimensional continued-fraction algorithms,J. Stat. Phys., this issue.
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Baldwin, P.R. A multidimensional continued fraction and some of its statistical properties. J Stat Phys 66, 1463–1505 (1992). https://doi.org/10.1007/BF01054430
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DOI: https://doi.org/10.1007/BF01054430