The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere

Transcription

1 The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere D M Hardcastle CONTENTS Introduction The Three-Dimensional Gauss Transformation Strategy of the Proof 4 Numerical Estimation 5 Numerical Results 6 Discussion References A proof that the three-dimensional Gauss algorithm is strongly convergent almost everywhere is given This algorithm is equivalent to Brun s algorithm and to the modified Jacobi-Perron algorithm considered by Podsypanin and Schweiger The proof involves the rigorous computer assisted estimation of the largest Lyapunov exponent of a cocycle associated to the algorithm To the best of my knowledge this is the first proof of almost everywhere strong convergence of a Jacobi-Perron type algorithm in dimension greater than two INTRODUCTION 000 AMS Subject Classification: Primary J70; Secondary K50 Keywords: Multidimensional continued fractions Brun s algorithm Jacobi-Perron algorithm strong convergence Lyapunov exponents Multidimensional continued fraction algorithms have been widely studied since the nineteenth century They were first introduced by Jacobi who considered a generalisation of the famous one-dimensional continued fraction algorithm to two dimensions [Jacobi 868] Jacobi s algorithm is now known as the Jacobi-Perron algorithm (JPA) in recognition of the fundamental work of Perron who defined the algorithm in arbitrary dimension and proved its convergence [Perron 907] Other famous algorithms include Brun s [Brun 57] Selmer s [Selmer 6] and the modified Jacobi-Perron algorithm considered by Podsypanin [Podsypanin 77] and Schweiger [Schweiger 79] For a given irrational vector (ω ω d ) [0 ] d \ Q d all multidimensional continued fraction algorithms produce a sequence of rational vectors (p (n)/q(n)p d (n)/q(n)) which converges to (ω ω d ) The algorithms mentioned above are known to be convergent in the weak sense ie lim n (ω p (n) ω d ) q(n) p d(n) =0 q(n) The approximations are said to be strongly convergent if lim n kq(n)(ω ω d ) (p (n)p d (n))k =0 ( ) c AKPetersLtd /00 $ 050 per page Experimental Mathematics : page

2 Experimental Mathematics Vol (00) No It is believed that the JPA the modified JPA and the algorithms of Brun and Selmer are strongly convergent almost everywhere ie for Lebesgue almost all (ω ω d ) [0 ] d \ Q d equation ( ) holds However at the present time the only rigorous proofs of strong convergence are for two-dimensional algorithms In fact strong convergence of the two-dimensional JPA is a consequence of the results of Paley and Ursell [Paley Ursell 0] This observation was first made by K Khanin [Khanin 9] (see also [Schweiger 96]) The modified JPA and Brun s algorithm have also been proven to be strongly convergent almost everywhere in two dimensions (see [Meester 98] and [Schratzberger 98]) In the early 990s the significance of the Lyapunov exponents for the convergence properties of multidimensional continued fraction algorithms was noted independently by D Kosygin [Kosygin 9] and P Baldwin [Baldwin 9a Baldwin 9b] A d-dimensional continued fraction algorithm has d Lyapunov exponents λ λ λ d The approximations produced by an algorithm are exponentially strongly convergent almost everywhere if and only if λ < 0 In [Ito et al 9] Ito Keane and Ohtsuki introduced a cocycle with largest Lyapunov exponent λ They used this cocycle to give a computer assisted proof of almost everywhere exponential strong convergence of the two-dimensional modified JPA In [Hardcastle Khanin 00] K Khanin and I introduced a method which in principle can be used to prove almost everywhere strong convergence in arbitrary dimension This scheme requires a good knowledge of the invariant measure of the endomorphism associated to the algorithm The method was applied to the ordered JPA which is equivalent to the modified JPA and Brun s algorithm The ordered JPA is particularly suitable for study by this scheme because the invariant density is known explicitly In [Hardcastle Khanin 0] the method of Ito et al was discussed in arbitrary dimension In particular it was explained how the problem of proving almost everywhere exponential strong convergence of the ordered JPA can be reduced to a finite number of calculations In this paper these calculations are carried out for the threedimensional ordered JPA This results in a computer assisted proof of almost everywhere strong convergence of the three-dimensional ordered JPA Note that I use the term proof here despite the fact that I do not attempt to control round-off errors I will leave the issue of whether the term proof is appropriate to the individual reader Both in [Hardcastle Khanin 0] and in this paper the ordered JPA is called the d-dimensional Gauss algorithm This name was introduced in [Hardcastle Khanin 0] where it was suggested that the ordered JPA should be considered to be the most natural generalisation of the one-dimensional Gauss transformation THE THREE-DIMENSIONAL GAUSS TRANSFORMATION This section begins with a description of the threedimensional Gauss algorithm The ergodic properties of the algorithm are then discussed and we explain the significance of the Lyapunov exponents for the convergence properties of the algorithm Let = {(ω ω ω ) [0 ] : ω ω ω } Define T : by T (ω ω ω ) ½ ¾ ω ω ω ω ω ½ ¾ ω = ω ω ω ω ω ω ½ ¾ ω ω ω ½ ¾ if > ω ; ω ω if ω ½ ¾ > > ω ; ω ω if ω ω > ω ½ ω ¾ ( ) In this formula {x} denotes the fractional part of a real number x ie {x} = x [x] where[x] denotes the integer part of x Definition The transformation T : is called the three-dimensional Gauss transformation Two numbers are naturally associated to the process of calculating T (ω) = T (ω ω ω ): the integer part of /ω which we denote by m(ω) ie m(ω) =[/ω ] and the position in which {/ω } =/ω m(ω) is placed whichwedenotebyj(ω) ie the j(ω) th coordinate of T (ω) is{/ω } Define (mj) = {ω : m(ω) =m j(ω) =j} It is easy to check that T (mj) is injective and the image of (mj) under T is the whole of The inverse of T (mj) which we denote T (mj) isgivenby T (mj) (ω ω ω ) m ω = m ω m ω ω m ω ω m ω ω m ω ω m ω ω m ω ω m ω if j =; if j =; if j =

3 Hardcastle: The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere For each (m j) N { } we define a matrix ea (mj) GL(4 ): m 0 0 m 0 0 ea (m) = ; A(m) e = ; m 0 0 ea (m) = The matrices A e (mj) give the action of T (mj) on rational vectors: T p (mj) q p q p q = ep eq ep eq ep eq eq q if and only if ep ep = A e (mj) p p ep p To form simultaneous rational approximations to a vector ω we consider the orbit of ω under T : ω T 7 T ω T 7 T 7 T n ω To this trajectory we associate the sequence (m j )(m n j n )where Define m i = m(t i ω) j i = j(t i ω) ec n (ω) = e A (m j ) e A (m j ) ea (mn j n ) ( ) The matrix C e n (ω) gives the n th approximation to ω More precisely let q(n 0) q(n ) q(n ) q(n ) ec n (ω) = p (n 0) p (n ) p (n ) p (n ) p (n 0) p (n ) p (n ) p (n ) p (n 0) p (n ) p (n ) p (n ) Denote p(n i) q(n i) = p (n i) q(n i) p (n i) q(n i) p (n i) 0 i q(n i) ( ) We will consider p(n 0)/q(n 0) as the n th approximation to ω Denote τ n = p(n 0) q(n 0) = p (n) q(n) p d(n) ( 4) q(n) Definition A sequence of rational vectors τ n = (p (n)/q(n)p d (n)/q(n)) is exponentially strongly convergent to ω if there exist constants k > 0 α > 0 such that kq(n)ω (p (n)p d (n))k kq(n) α The purpose of this paper is to prove that the threedimensional Gauss algorithm is exponentially strongly convergent almost everywhere ie for almost all ω the sequence τ n defined by ( 4) is exponentially strongly convergent to ω It will be convenient to present ( ) in a slightly different fashion Let A (mj) =( A e (mj) ) t where A t denotes the transpose of a matrix A Define a matrix-valued function on by Denote Then A(ω) =A (m(ω)j(ω)) C n (ω) =A(T n ω) A(T ω)a(ω) C n (ω) =( e C n (ω)) t = A (mnj n) A (mj )A (mj ) We now discuss the ergodic properties of the map T Define ρ(ω) = X ω π S π() ω π() ω π() ω π() ω π() ω π() = ω ω ω ( ω )( ω ω ) ( ω )( ω ω ) ( ω )( ω ω ) ( ω )( ω ω ) ( ω )( ω ω ) ( 5) ( ω )( ω ω ) where S is the group of permutations of { } Let K = R ρ(ω) dω It is easy to check that the probability measure defined by (X) = ρ(ω) dω X a Borel subset of d K X is T -invariant and ergodic (see [Schweiger 79]) This measure is the unique absolutely continuous T -invariant probability measure

4 4 Experimental Mathematics Vol (00) No The endomorphism T the matrix-valued function A and the invariant measure together form a cocycle whichwedenote(ta) This cocycle is integrable ie log(max(ka(ω)k )) (dω) < Let λ λ λ λ 4 be the Lyapunov exponents of the cocycle (TA) (see [Oseledets 68]) The following result relates the convergence properties of the algorithm to its Lyapunov exponents Theorem (i) The largest Lyapunov exponent λ is strictly positive and simple (ii) For almost all ω lim n n log q(n) =λ (iii) The sequence τ n is exponentially strongly convergent to ω for almost all ω if and only if λ < 0 This theorem which is based on the work of Lagarias [Lagarias 9] was proved in [Hardcastle Khanin 00] STRATEGY OF THE PROOF By Theorem in order to prove almost everywhere strong convergence it is enough to show that λ < 0 It is difficult to estimate the exponent λ directly so we consider a new cocycle (TD) whose largest Lyapunov exponent λ (D) isλ Our strategy will be to estimate λ (D) by use of the Subadditive Ergodic Theorem Define ω ω ω D = 0 0 ; D = 0 0 and 0 0 D = 0 0 ω ω ω Let D(ω) =D j(ω) Alsodefine 0 0 ω ω ω 0 0 D n (ω) =D(T n ω) D(T ω)d(ω) Let λ (D) denote the largest Lyapunov exponent of the cocycle (TD) ; Lemma λ (D) =λ In [Hardcastle Khanin 0] we gave a proof of this lemma and a description of the construction of the matrices D(ω) Ourconstructionwasbasedonanobservation made by Lagarias [Lagarias 9] but matrices similar to D(ω) have also been considered by Ito Keane and Ohtsuki [Ito et al 9 Ito et al 96] The following lemma is an immediate consequence of the Subadditive Ergodic Theorem [Kingman 68] R Lemma λ (D) =inf n N log kd n n (ω)k (dω) Combining Lemmas and and Theorem we have: Theorem The three-dimensional Gauss algorithm is exponentially strongly convergent almost everywhere if and only if there exists n N such that log kd n (ω)k (dω) < 0 ( ) n We will give a computer assisted proof that condition ( ) holds for n = 8 The following results which were proved in [Hardcastle Khanin 0] will be useful when we estimate the integral in ( ) We firstshowthatthematrixd n (ω) can be expressed in terms of ω and its approximations Proposition 4 For all n N D n(ω) = q(n )ω p (n ) q(n )ω p (n ) q(n )ω p (n ) q(n )ω p (n ) q(n )ω p (n ) q(n )ω p (n ) q(n )ω p (n ) q(n )ω p (n ) q(n )ω p (n ) Recall that in Section we defined (mj) = {ω : m(ω) =m j(ω) =j} These sets form a Markov partition for T Let (mj )(m nj n) denote an element of the n th level of the Markov partition: (mj )(m nj n) = {ω : m(t i ω) = m i j(t i ω)=j i for i n} The following proposition describes how the vertices of the simplex (m j )(m n j n ) can be calculated

5 Hardcastle: The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere 5 Proposition 5 Let where V (m j )(m n j n ) =(v il ) il 4 = e A (m j ) e A (m j ) ea (mn j n )V 0 V 0 = Then the vertices of (mj )(m nj n) are v i = vi v i v 4i i 4 v i v i v i The next result explains how the maximum value of kd n (ω)k over a simplex can be calculated Let k k denote an arbitrary norm on R Then the corresponding norm of a linear operator D : R R is given by kdk = kdvk sup v R \{0} kvk Proposition 6 The maximum value of log kd n (ω)k over a simplex (m j )(m n j n ) occurs at a vertex of ie max log kd n(ω)k = max log kd n(v i )k ω i 4 where v v 4 are the vertices of Proof: See Corollary 47 of [Hardcastle Khanin 0] We now consider how we can calculate an upper and a lower bound for the density ρ over a simplex Proposition 7 The maximum value of ρ over a simplex occurs at one of the vertices of ie where v is a vertex of max ρ(ω) =ρ(v) ω Proof: See Corollary 4 of [Hardcastle Khanin 0] Corollary 8 If is a simplex then ( ) max K ρ(v i) vol( ) i 4 where v v 4 are the vertices of and vol denotes three-dimensional Lebesgue measure Define functions s s s s s s s : R by s (ω) =ω s (ω) =ω s (ω) =ω s (ω) =ω ω s (ω) =ω ω s (ω) =ω ω s (ω) =ω ω ω Then ρ(ω) = s (ω) s (ω) s (ω) s (ω) s (ω) s (ω) s (ω) s (ω) s (ω) s (ω) Lemma 9 Let be a simplex with vertices v v 4 The maximum value of each function s s s is attained at a vertex of ie max s l(ω) = max s l(v i ) ω i 4 where l = Let s = max i 4 s (v i ) and similarly for s s s Corollary 0 For ω ρ(ω) s s s s s s s s s s Let the above lower bound for ρ(ω) be denoted ρ( ) Then we have: Corollary ( ) K ρ( )vol( ) The following two lemmas will allow us to obtain a global bound for kd n (ω)k Lemma For any ω kd(t ω)d(ω)k 7 Proof: Suppose j(ω) = sothat T (ω) =T (ω ω ω )= m ω ω ω ω ω

6 6 Experimental Mathematics Vol (00) No If j(t ω) =then D(T ω)d(ω) m ω ω ω ω ω ω ω ω = m ω m ω m ω = 0 0 ( ) 0 0 Suppose kxk = k(x x x )k = p x x x = Then kd(t ω)d(ω)xk m ω m ω m ω x = 0 0 x 0 0 x = ( m ω )x m ω x m ω x x x ( m ω ) m ω m ω x x x x =4 since 0 ω ω ω m ThuskD(Tω)D(ω)k Now if j(t ω) =orthend(t ω)d(ω) will be the same as ( ) up to a change in the order of the rows Thus for any ω such that j(ω) =kd(t ω)d(ω)k Now suppose that j(ω) = sothat ω T (ω) =T (ω ω ω )= m ω ω ω ω Then if j(t ω) = D(T ω)d(ω) ω ω m ω ω ω 0 0 = 0 0 ω ω ω m ω m ω m ω = ω ω ω 0 0 So if x x x =then kd(t ω)d(ω)xk m ω m ω m ω = ω ω ω 0 0 x x x = ( m ω )x m ω x m ω x (ω x ω x ω x )x ( m ω ) m ω m ω x x x (ω ω ω ) x x =7 Thus kd(t ω)d(ω)k 7 If j(t ω) = or then the same estimate holds and it can easily be checked that if j(ω) = then the same estimate is valid Lemma For any ω kd(ω)k = q ω ω ω (ω ω ω ) 4ω Proof: This is an easy calculation Corollary 4 If m(ω) m then kd(ω)k Proof: By the lemma m r 9m 4 4m kd(ω)k = ω ω ω qω 4 ω4 ω4 (ω ω ω ω ω ω ω ω ω ) If m(ω) m then ω ω ω m so s kd(ω)k m m 4 m 4 m = m r 9m 4m 4 4 NUMERICAL ESTIMATION As explained in the previous section to prove that the three-dimensional Gauss algorithm is exponentially strongly convergent almost everywhere it is enough to show that I n := log kd n (ω)k (dω) < 0 n for some n N In this section we will discuss how an upper bound for the integral I n can be found Choice of norm The integral I n obviously depends on which norm on the space of linear maps R R is used Any norm which is defined by kdk = kdvk sup v R \{0} kvk (4 )

7 Hardcastle: The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere 7 for some norm k k on R can be used In our calculations we will use the norm defined by (4 ) where k k is the standard Euclidean norm on R ie kvk = k(v v v )k = p v v v Recall that the norm of a linear map D : R R induced by the Euclidean norm on R is given by kdk =max{γ : γ is an eigenvalue of D t D} This norm has the disadvantage that it is hard to compute in comparison to many other norms Nevertheless this norm will be used in our calculations because Monte- Carlo integration shows that the value of n required to obtain log kd n (ω)k (dω) < 0 using the Euclidean norm is smaller than the value needed using any of the other standard norms such as kvk = P i= v i or kvk =max i v i The advantage of using a smaller value of n outweighs the disadvantage that the norm is harder to compute Choice of n Firstly it is necessary to find n such that I n < 0 This can easily be done by estimating I n non-rigorously using a Monte-Carlo type technique There are two conflicting factors which determine the choice of n On one hand it is obviously desirable to have n as small as possible On the other hand the larger I n is in absolute value the larger our error terms in the estimation procedure can be In practice a compromise has to be reached between these two factors Table contains Monte-Carlo estimates for the integrals I n n I n TABLE TheapproximatevalueoftheintegralI n for n = 9 Our calculation will show that I 8 < 0sincen = 8 seems to be the best compromise Constant factors Recall that the invariant measure is given by (dω) = ρ(ω) K dω Thus I 8 = log kd 8 (ω)kρ(ω) dω 8K Let γ(d) denote the largest eigenvalue of D t DThen I 8 = log p γ(d 8 (ω))ρ(ω) dω 8K = log γ(d 8 (ω)) ρ(ω) dω 6K We now describe how the integral bi 8 = log γ(d 8 (ω)) ρ(ω) dω can be estimated 4 Partitioning of We will estimate the integral bi 8 by considering as the union of the simplices (mj )(m 8j 8) and estimating the integral over each of these simplices separately Of course there are infinitely many simplices (mj )(m 8j 8) so the set of these simplices is split into two parts: a finite part where we consider each simplex individually and an infinite part where we use a crude upper bound for the integral Let Ξ 8 denote the set of elements of the 8 th level of the Markov partition ie Ξ 8 = { (mj )(m 8j 8) : m i N and j i for i 8} The finite part of Ξ 8 which is used for integration is denoted 5 Thechoiceof The set Ω = [ should have as large a measure as possible Consequently the elements of should be chosen to be as large in measure as possible There are two general principles which can be used in the choice of Firstly if all the m i are relatively small then the measure of (mj )(m 8j 8) will be relatively large Secondly if two or more m i are large then the measure will be small

8 8 Experimental Mathematics Vol (00) No 6 Splitting of the elements of The set is divided into three parts = () () () according to the relative measures of the simplices The simplices in () are of relatively small measure; those in () are of relatively large measure while () contains simplices of intermediate measure Since the integral over a simplex will be estimated in terms of the values of the integrand at the vertices it follows that in general the larger the simplex the less accurate the estimate will be Consequently the simplices in () and () will be split into subsimplices and the integral over each of these subsimplices will be estimated individually Note that the vertices of a simplex can be calculated using Proposition 5 Asimplex () with vertices v v 4 is divided into four pieces in the following way Define Then c = 4 (v v v v 4 ) = 4[ i= where is the simplex with vertices v v v c; has vertices v v v 4 c; has vertices v v v 4 c; and 4 has vertices v v v 4 c LetS 4 ( ) ={ 4 } The simplices in () are divided into 6 pieces Firstly () is split into four pieces in the same way as described above Each of these four pieces is split into four pieces in the same way Thus if () let S 4 ( ) ={ 4 } Then if S 6 ( ) denotes the 6 subsimplices that is split into we have Let Then S 6 ( ) = i 4[ S 4 ( i ) i= [ [ = () S 4 ( ) S 6 ( ) () () Ω = [ 7 Estimation of the integral over and the measure of Let v v 4 be the vertices of (i) For i =4 calculate the matrix D 8 (v i )using Proposition 4 (ii) Calculate d n ( ) =log max γ(d 8(v i )) i 4 (iii) Find ρ(v i )using( 5)andletρ( ) = max i 4 ρ(v i) (iv) Calculate a lower bound for the density over using Corollary 0 (v) Find the volume of using vol( ) = v v 6 det v v v 4 v (vi) Let I (u) ( ) = ( d n ( )ρ( )vol( ) if d n ( ) < 0; d n ( )ρ( )vol( ) if d n ( ) > 0 Notice that log γ(d 8 (ω)) ρ(ω) dω I (u) ( ) The symbol (u) is intended to denote that this is an upper bound for the unnormalised integral Note that 8 log kd 8 (ω)k (dω) 6K I(u) ( ) (vii) Let (u) ( ) =ρ( )vol( ) Then ρ(ω) dω (u) ( ) 8 Estimation of the integral over Ω and the measure of Ω The following estimates hold: log γ(d 8 (ω)) ρ(ω) dω X I (u) ( ) and Ω Ω ρ(ω) dω X (u) ( ) 9 An upper bound for the integral over the complementary set \ Ω By Lemma kd(t ω)d(ω)k 7 Thus kd 8 (ω)k ( 7) 4 = 49 which implies that \Ω log kd 8 (ω)k (dω) log(49)( \ Ω) = log(49) (Ω) log(49) X K (u) ( )

9 Hardcastle: The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere 9 Set of simplices All simplices (mj )(m 8j 8) for which j i fori =8 and: #{i : m i } =8or#{i : m i } =7and#{i :4 m i 6} = #{i :7 m i 4} = #{i : m i } =7 #{i :5 m i } = #{i : m i } =7 4 #{i :4 m i 6} = #{i : m i } =6 5 #{i :4 m i 6} = #{i : m i } =5 6 #{i :4 m i 6} =4 #{i : m i } =4 7 #{i :4 m i 6} =5 #{i : m i } = 8 #{i :4 m i 6} =6 #{i : m i } = 9 #{i :4 m i 6} =7 #{i : m i } = 0 #{i :4 m i 6} =8 #{i :4 m i 6} = #{i :7 m i 9} = #{i : m i } =6 TABLE Thedefinition of 0 An upper bound for I 8 We have the following final upper bound for I 8 : I 8 = log kd 8 (ω)k (dω) 8 = log γ(d 8(ω)) ρ(ω) dω 6K Ω log kd 8 (ω)k (dω) 8 \Ω X I (u) ( ) 4 6K log(7) X (u) ( ) K 5 NUMERICAL RESULTS The method described in the previous section can easily be implemented on a computer In this section we will present numerical results which were obtained using this method and which prove that I 8 = log kd 8 (ω)k (dω) < 0 8 For convenience of presentation of these results we split the set of simplices which are used for integration into parts These sets are definedintable The simplices in were split into 6 pieces those in were split into 4 pieces and the remaining simplices were not split at all ie Let () = () = () = Ω i = [ i [ i= i Table contains the numerical results The program which produced these results was written in C To evaluate γ(d 8 (ω)) which requires the calculation of the largest eigenvalue of the symmetric matrix (D 8 (ω)) t D 8 (ω) the routines tred and tqli (see pages 474 and 480 of [Press et al 9]) were used i Upper bound for Lower bound for the integral over Ω i the measure of Ω i ie 6K P i I (u) P ( ) ie K i (u) ( ) Total TABLE Results for regions Ω Ω The first of these routines transforms a real symmetric matrix into tridiagonal form by a series of orthogonal transformations The second routine finds the eigenvalues of such a matrix (Quicker algorithms to do this may exist; a referee suggested that it might be better for future work to use the singular value decomposition to

10 40 Experimental Mathematics Vol (00) No compute the norm rather than computing the eigenvalues and noted that there is a variety of public domain software for both eigenvalues and singular values notably LAPACK) The evaluation of the matrix V n = ea (m j ) e A (m j ) ea (mn j n )V 0 (see Proposition 5) was performed using integer variables All other calculations were performed to double precision The program took approximately 750 hours on a Sun Ultra 5 to produce the given results The program was also written in the Matlab language using the built-in eigenvalue and matrix multiplication routines Many of the calculations were repeated using this program and the results agreed with the ones obtained by the C program to the accuracy stated Proposition 5 For any ω \ Ω 8 log kd 8(ω)k (5 ) Proof: Since the set Ω contains all simplices (m j )(m 8 j 8 ) for which m i 6for i 8 if ω \ Ω then for some l 8 m l 7 Consequently by Corollary 4 kd(t l ω)k 7 r Also since m(ω) foranyω for i 8 kd(t i ω)k 4 4 (5 ) The matrix product D(T 7 ω) D(T ω)d(ω) can be viewed as the product of three matrices of the form D(T i ω)d(t i ω) one matrix D(T i ω) for which (5 ) holds and the matrix D(T l ω) Thus by Lemma kd 8 (ω)k ( 7) This implies (5 ) 7 r Theorem 5 log kd 8 (ω)k (dω) Proof: By Table and Proposition 5 log kd 8 (ω)k (dω) ( ) = Corollary 5 The three-dimensional Gauss algorithm is exponentially strongly convergent almost everywhere 6 DISCUSSION Similar calculations to those described above can be carried out in any dimension However the number of calculations necessary increases very rapidly with the dimension One reason for this is that the value of n necessary to have I n < 0 increases with the dimension In dimension I n < 0forn 5 whereas in four dimensions I n < 0forn This obviously increases the number of calculations necessary Another problem is that λ (D) 0 as the dimension increases This means that the error terms must be smaller so more calculations will be necessary The calculation can of course be broken down into several disjoint calculations which can then be carried out simultaneously on different computers More modern machines than the one used in this paper may be of the order of 0 times quicker Using several of these machines would result in a 00 times improvement in the speed of the calculation However even allowing for such an improvement I believe that the calculations necessary to prove strong convergence in dimension 4 would still take a great deal longer than the calculations described here There seems to be no reason to doubt that the result is true in all dimensions It should be possible to prove strong convergence of other three-dimensional continued fraction algorithms by similar methods to those in this paper if the invariant density is known explicitly If a good approximation to the invariant density is known then it should also be possible to prove the results However obtaining such an approximation seems to be very difficult ACKNOWLEDGMENTS This paper is the natural continuation of a series of papers by myself and K Khanin [Hardcastle Khanin 00] [Hardcastle Khanin 0] [Hardcastle Khanin 0] I am very grateful to K Khanin for numerous useful discussions concerning the work in this paper I am also grateful to R Khanin for advice about computational matters I also thank the Engineering and Physical Sciences Research Council of the UK for financial support

11 Hardcastle: The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere 4 REFERENCES [Baldwin 9a] P R Baldwin A multidimensional continued fraction and some of its statistical properties Jour Stat Phys 66 (99) [Baldwin 9b] P R Baldwin A convergence exponent for multidimensional continued-fraction algorithms Jour Stat Phys 66 (99) [Brun 57] V Brun Algorithmes euclidiens pour trois et quatre nombres In ième Congre Math Scand Helsinki (957) [Itoetal96]TFujitaSItoMKeaneandMOhtsuki On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof Ergod Th and Dyn Sys 6 (996) 45 5 [Hardcastle Khanin 00] D M Hardcastle and K Khanin On almost everywhere strong convergence of multidimensional continued fraction algorithms Ergod Th and Dyn Sys 0 (000) 7 7 [Hardcastle Khanin 0] D M Hardcastle and K Khanin Continued fractions and the d-dimensional Gauss transformation Commun Math Phys 5 (00) [Hardcastle Khanin 0] D M Hardcastle and K Khanin The d-dimensional Gauss transformation: strong convergence and Lyapunov exponents Experimental Mathematics : (00) 9 9 [Ito et al 9] S Ito M Keane and M Ohtsuki Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm Ergod Th and Dyn Sys (99) 9 4 [Jacobi 868] C G J Jacobi Allgemeine Theorie der kettenbruchähnlichen Algorithmen in welchen jede ahl aus drei vorhergehenden gebildet wird J Reine Angew Math 69 (868) 9 64 [Khanin 9] K Khanin Talk at the International Workshop on Dynamical Systems Porto (99) [Kosygin 9] D V Kosygin Multidimensional KAM theory from the renormalization group viewpoint In Dynamical Systems and Statistical Mechanics (Ed: Ya G Sinai) Advances in Soviet Mathematics (99) 99 9 [Kingman 68] J F C Kingman The ergodic theory of subadditive stochastic processes J Royal Stat Soc B 0 (968) [Lagarias 9] J C Lagarias The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms Mh Math 5 (99) 99 8 [Meester 98] R Meester A simple proof of the exponential convergence of the modified Jacobi-Perron algorithm Ergod Th and Dyn Sys 9 (999) [Oseledets 68] V I Oseledets A multiplicative ergodic theorem Liapunov characteristic numbers for dynamical systems Trans Moscow Math Soc 9 (968) 97 [Paley Ursell 0] R E A C Paley and H D Ursell Continued fractions in several dimensions Proc Cambridge Philos Soc 6 (90) 7 44 [Perron 907] O Perron Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus Math Ann 64 (907) 76 [Podsypanin 77] E V Podsypanin A generalization of the algorithm for continued fractions related to the algorithm of Viggo Brun ap Naucn Sem Leningrad Otdel Mat Inst Steklov 67 (977) English translation: Journal of Soviet Math 6 (98) [Press et al 9] W H Press S A Teukolsky W T VetterlingandBPFlanneryNumerical Recipes in C: The Art of Scientific Computing Second Edition Cambridge University Press (99) [Schratzberger 98] B R Schratzberger The exponent of convergence for Brun s algorithm in two dimensions Sitzungsber Abt II 07 (998) 9 8 [Schweiger 79] F Schweiger A modified Jacobi-Perron algorithm with explicitly given invariant measure In Ergodic Theory Proceedings Oberwolfach Germany 978 Lecture Notes in Mathematics 79 (979) 99 0 Springer-Verlag [Schweiger 96] F Schweiger The exponent of convergence for the -dimensional Jacobi-Perron algorithm In Proceedings of the Conference on Analytic and Elementary Number Theory in Vienna 996 (Ed: W G Nowak and J Schoissengeier) 07 [Selmer 6] E Selmer Om flerdimensjonal Kjede brøk Nordisk Mat Tidskr 9 (96) 7 4 D M Hardcastle Department of Mathematics Heriot-Watt University Edinburgh EH4 4AS United Kingdom (DMHardcastle@mahwacuk) Received February 8 00; accepted in revised form August 4 00