A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM

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1 MATHEMATICS OF COMPUTATION Volume 78 Number 268 October 2009 Pages S (09) Article electronically published on January A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI Abstract. It has been believed that the continued fraction expansion of ( ) (1 is a Q-basis of a real cubic field) obtained by the modified Jacobi- Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of ( ) ( x denoting the fractional part of x). We present a new algorithm which is something like the modified Jacobi-Perron algorithm and give some experimental results with this new algorithm. From our experiments we can expect that the expansion of ( ) with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields. 1. Introduction The study of continued fractions has a long history dating back to J. Wallis ( ) and Ch. Huygens ( ) [8]. In particular many kinds of higherdimensional continued fractions have been studied starting with K.G. Jacobi ( ) [7]. A central problem has been to find a higher-dimensional generalization of Legendre s theorem concerning the periodic continued fractions. In fact the following conjecture has been believed. Conjecture. Let s be a Q-basis of real number field K with [K : Q] = s +1. Then the expansion of ( 1... s ) by the Jacobi-Perron algorithm is eventually periodic; cf. [11]. Although Bernstein [1] gave some classes of periodic continued fractions obtained by the Jacobi-Perron algorithm the conjecture is still open. For example the periodicity of the Jacobi-Perron algorithm for ( ) has not been established (see [2] [12]). We also gave a numerical experiments (cf. Table B Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of ( ) by the modified Jacobi-Perron algorithm. In this paper we give some candidates of algorithms of continued fraction expansion of dimensions 2 and 3 which can be easily generalized to any dimension. By numerical experiments we checked that for instance ( 3 m 3 m 2 )(2 m 5000m Z 3 m / Q) and( 4 m 4 m 2 4 m 3 )(2 m 5000m Z m/ Q) obtained by our algorithm become periodic. We showed the periodicity Received by the editor May and in revised form August Mathematics Subject Classification. Primary 11J70; Secondary 68W25. Key words and phrases. Diophantine approximation multidimensional continued fraction algorithm c 2009 American Mathematical Society Reverts to public domain 28 years from publication

2 2210 JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI of the expansion of ( ) K 2 by our algorithm for small classes of cubic fields K including totally real cases and pure cubic cases; cf. Theorems The algorithms given in this paper are motivated by the algorithms given in [12] which are quite different from the Jacobi-Perron algorithm but our algorithms are related to the so-called modified Jacobi-Perron algorithm; cf. [3] [4] [5] [10]. 2. The cubic case In this section we consider real cubic fields K (including totally real cases and not totally real cases). We denote by X K the set defined by X K := {( ) K 2 1 are linearly independent over Q} I 2 where I =[0 1). We define the transformation T K on X K by T K ( ) := ( 1 1 ) if > N() ( 1 1 ) if < N() N() N() for ( ) X K where x is the floor function of x and N(x) isthenormofx K over Q. Lemma 2.1. The transformation T K is well defined. Proof. Let ( ) X K. It suffices to show that = N(). Then we have = N() N() N(). N() N() / Q. Hence independent over Q sothat and X K. N() N() N() N(). We suppose N() Since and are linearly is a quadratic irrational N() K which is a contradiction. It is easy to see that T K ( ) We define the integer-valued functions a b and e on X K as follows: 1 if > N() N() a( ) := if < N() N() if > N() N() b( ) := 1 if < N() N() 0 if > N() N() e( ) := 1 if < N() for ( ) X K. N()

3 A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM 2211 We put (a n b n e n )=(a n ( )b n ( )e n ( )) := (a(t n 1 n 1 n 1 K ( )b(tk ( )e(tk ( )) (n Z >0) S( ) :={(a n ( )b n ( )e n ( ))} n=1. The sequence S( ) will be referred to as the expansion of ( ) X K by T K ; T K gives rise to a 2-dimensional continued fraction expansion which will be called the Algebraic Jacobi-Perron Algorithm(AJPA). Throughout our paper n n are numbers defined by ( n n ):=TK( n ) (n Z 0 ). Notice that (a n b n e n ) Z 0 Z 0 {0 1} (n Z 0 ). For each (a b e ) Z >0 Z 0 {0 1} we put b if e =0 1 0 a (2.1) A (a b e ) := 1 0 a if e =1 0 1 b M n ( ) = p n( ) p n( ) p n ( ) (2.2) q n( ) q n( ) q n ( ) := A (a1 r n( ) r n( b 1 e 1 ) A (an b n e n ). ) r n ( ) Definition. We denote by P K the set {( ) X K there exist m n Z >0 such that m n and TK m ( ) =TK( n )}. If ( ) P K the expansion S( ) byt K becomes periodic and vice versa. In what follows we mean by the period the period obtained by choosing the shortest period and preperiod. For the periodic continued fraction obtained by AJPA we have the following proposition. In a way similar to Perron [9] we can show Proposition 2.2. Let ( ) P K. Then there exists a constant c( ) > 0 and η( ) > 0 such that η( ) 3 2 and p n r n q n r n c( ) r η() n c( ) r η() n holds. Furthermore η( ) = 3 2 holds if and only if K is not a totally real cubic field. Remark 2.3. (1) Based on our many experiments (cf. Tables A C D) we can hope that X K = P K.

4 2212 JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI (2) In view of Proposition 2.2 we see that ( ) ( ) S ( ) S( ) for ( ) ( ) P K. Bernstein [1] gave some classes of periodic continued fractions obtained by the Jacobi-Perron algorithm. We can also give some examples of periodic expansions obtained by the AJPA for example: Theorem 2.4. Let K = Q( 3 m 3 +1) with m Z >0. Let ( ) =( 3 m 3 +1 m 3 (m 3 +1) 2 m 2 ).Then( ) P K and the length of the period is 2. Proof. It is easy to see that ( ) X K. We have N() =1and N() = (m 3 +1) 2 m 6 =2m First we consider the case where m 2. Since 2m 3 +1> 3 m 3 +1+m we have > sothate 1 = 0. Therefore we get N() N() ( 1 1 )=T K ( ) =( 1 a 1 b 1) =( 3 (m 3 +1) 2 + m 3 m 3 +1+m 2 3m 2 3 m 3 +1+m 2m) =( 3 (m 3 +1) 2 + m 3 m m 2 3 m 3 +1 m). We see that N( 1 ) =9m 3 +1and N( 1 ) = 1. One can see that 3 (m3 +1) 2 + m 3 m m 2 < 3 m 3 +1 m. 9m3 +1 Therefore we have e 2 =1. Thusweget ( 2 2 )=T K ( 1 1 )=( 1 a 2 ) 1 b 2 ) 1 1 =( 3 m m 3m 3 (m 3 +1) 2 + m 3 m 3 +1+m 2 3m 2 ) =( 3 m 3 +1 m 3 (m 3 +1) 2 + m 3 m m 2 ) =( 1 1 ). Therefore we have ( 2 2 )=( 1 1 ). Hence {TK n ( )} n=0 is periodic with 2 as the length of its (shortest) period. Thus we get n a n 3m 2 3m 3m 2 b n 2m 3m 2 3m e n a n = a n 2 b n = b n 2 and e n = e n 2 for all n 4. Second we consider the case m = 1 i.e. ( 0 0 )=( ). Then we see that ( 1 1 )=( ) ( 2 2 )=( 0 0 ). We get

5 A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM 2213 n 1 2 a n 0 2 b n 1 1 e n 1 0 a n = a n 2 b n = b n 2 and e n = e n 2 for all n 3. We put for p q (p q Z are coprime) dh( p q ):=max{ log 10 p +1 log 10 q +1 } dh(0) := 0. The function dh can be extended to Q[x]: for g(x) = dh(g) := max {dh(a i)}. 0 i n Furthermore we put for ( ) K 2 n a i x i Q[x] put i=0 dh( ) :=max{dh(mpol()) dh(mpol())} where mpol(γ) Q[x] (γ K) is the monic minimal polynomial of γ. We put for n Z 0 and ( ) X K dh AJPA (n; ) :=dh( n n ) rdh AJPA (n; ) := dh( n n ) dh( 0 0 ) where ( n n ):=TK n ( ). The function dh AJPA(n; ) (resp. rdh AJPA (n; )) is referred to as the nth decimal height of ( ) (resp.the nth relative decimal height of ( )) with respect to the AJPA. Let K = Q( 3 m) with m Z >0 and 3 m / Q. Wecomputedthelength of the periods of the expansion S( 3 m 3 m 2 ) for all m with 2 m 5000 ( 3 m/ Q) and these decimal heights; cf. Table A. For the calculation of the tables we used a computer equipped with GiNaC [6] on GNU C++. We confirmed that ( 3 m 3 m 2 ) P K for all m with 2 m 5000( 3 m/ Q). There are no reports on the periodicity of any of these pairs of numbers obtained by the Jacobi-Perron algorithm or any modified Jacobi-Perron algorithms except for the algorithms of [12]. In [2] Elsner and Hasse gave numerical results for 36 pairs of cubic numbers with the Jacobi-Perron algorithm. They found 14 cases of periodicity but no sign of periodicity for the other 22 cases. In [12] the first author computed the dh(t n ( 3 m 3 m 2 )forsomen and m where T is the transformation related to the Jacobi-Perron algorithm. Tamura observed that it becomes gradually bigger and bigger as a function of n for many 2 m 104. This suggests that for some m lim dh(t n ( 3 m 3 m 2 )= n i.e. an explosion of the size of minimal polynomials. We computed dh( T n ( ) for 0 n where the transformation T which is associated with the modified Jacobi-Perron algorithm is defined by the following:

6 2214 JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI For (x y) [0 1) 2 (1xy are linearly independent over Q) ( y x 1 x 1 ) if x>y x T (x y) := ( 1 y 1 y x ) if x<y; y cf. Podsypanin [10]. This algorithm has been studied in connection with simultaneous diophantine approximation [3] [4] and [5]. Table B gives dh( T n ( ) for 0 n The table also suggests an explosion phenomenon related to the modified Jacobi-Perron algorithm. Theorem 2.5. Let δ m be the root of x 3 mx +1=0(m Zm 3) determined by 0 <δ m < 1. ThenK = Q(δ m ) is a cubic number field and (δ m δ 2 m) P K. Proof. The irreduciblity of g := x 3 mx + 1 is clear. It is also clear that g has a unique root δ m in [0 1). We consider TK n ( ) with( ) =(δ mδm). 2 Since N(δ m ) = N(δm) 2 δ =1and0<δ m < 1 we have m N(δm ) δ m 2 N(δ 2 m ) so that e 1 =0and ( 1 1 )=T K ( ) =( 1 a( ) b( )) Therefore =(m δm 2 (m 1)δ m ) =(1 δmδ 2 m ). 1 N(1 ) = 1 N(1 ) = δ m. 1 δ 2 m N(1 δ 2 m ) = 1 δ 2 m m(m 2) One can check that 1 δ 2 m m(m 2) >δ m if and only if (1 δ 2 m )2 >m(m 2)δ 2 m. Since (1 δm) 2 2 m(m 2)δm 2 =2(m 2)δm 2 +2(δm 2 δm)+δ 3 m 4 δm 6 > 0 we get 1 N(1 ) > 1 N(1 ). Therefore e 2 =0and ( 2 2 )=T K ( 1 1 )=( 1 a( 1 1 ) 1 b( 1 1 )) 1 ( 1 (m 1)δ 2 = m δ m + m 2 ) 2m +1 m 2 1 δ2 m (m 1)δ m +(m 1) 2m m 2 2m ( ) (m 1)δ 2 = m δ m +1 m 2 δ2 m (m 1)δ m +(m 1) 2m m 2. 2m 1 In view of N( 2 )= m(m 2) and N( 1 2)= m(m 2) weget 2 N(2 ) = δ2 m(m 2) 1 δm 2 < δ m(m 2) 2 1 δm 2 = N(2 ).

7 A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM 2215 Thus we get e 3 =1and ( 3 3 )=T K ( 2 2 )=( 2 a( 2 2 ) 1 b( 2 2 )) 2 2 =(δ m δm 2 δ m + m (m 1)) =(δ m δm 2 δ m +1). Since N( 3 ) = m 2 4m +4 3 < 3 N(3 ) N(3 ) holds if and only if 3 N(3 ) 3 N(3 )=( δ 2 m δ m +1) δ m (m 2) > 0 which is equivalent to (2.3) δ 4 m + δ 2 m 2δ m +1> 0. The inequality (2.3) holds since 0 <δ m < 1 2 which can be easily seen. Therefore we get e 4 = 1 and we have ( 4 4 )=T K ( 3 3 )=( 3 a( 3 3 ) 1 b( 3 3 )) 3 3 ( ) δm +1 = m 2 δ2 m + m 1 1 m 2 ( ) δm +1 = m 2 δ2 m +1 m 2 and N( 4 ) = 1 (m 2) 2 and N( 4 ) = m (m 2) 2 4. Therefore we see e 5 =0and N(4 ) follows. Thus we have 4 N(4 ) > ( 5 5 )=T K ( 4 4 )=( 1 a( 4 4 ) 4 b( 4 4 )) 4 4 =( δm 2 δ m + m 1 (m 2)δ m +1 1) =( δm 2 δ m +1δ m ) which implies ( 6 6 )=( δ2 m +1 m 2 δ m+1 m 2 )and( 7 7 )=( 3 3 ). Thus we obtain ( 3+4j 3+4j )=( 3 3 ) for all integer j 0. Thus we get n a n m m b n 0 0 m m 2 e n a n = a n 4 b n = b n 4 and e n = e n 4 for all n 8. We also confirmed that ( τ m τ 2 m ) P K for all integers m with 2 m 5000 where τ m is the maximal root of x 3 mx + 1 while the length of the (shortest) period is very long in some cases. Table C gives these results.

8 2216 JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI 3. The quartic case Let K be a real quartic field over Q. In this section we mean by X K and X K the sets defined by X K := {( ) K are linearly independent over Q} I 3 X K := {( ) X K there exists i {1 2 3} such that K = Q( i )}. For x K I we define φ(x) by φ(x) := { x 3 N(x) 1 if K = Q(x) if K Q(x). In a similar manner to the cubic case one can show the following. Lemma 3.1. Let ( ) X K. If φ( i )φ( j ) > 0 and φ( i )=φ( j ) for integers i and j with 1 i j 3 theni = j. For =( ) X K we define ρ() and ρ() =max{φ( i ) i {1 2 3}}. For =( ) X K from Lemma 3.1 it follows that {i {1 2 3} ρ() =φ( i )} =1 and we denote by ω() the uniquely determined number i {1 2 3} with ρ() = φ( i ). We define the transformation T K on X K as follows: For =( ) X K T K () =( ) with { 1 i 1 i if i = ω() i := i ω() i ω() if i ω() (i =1 2 3). We define P K in a similar fashion in Section 2 that is P K := { X K there exist m n Z >0 such that m n and T n K () =T m K ()}. We also define dh dh AJPA and rdh AJPA in a similar manner to Section 2 namely for =( ) X K and n Z 0 dh() := max i {123} {dh(p4/deg(p i) i )} dh AJPA (n; ) :=dh(tk()) n rdh AJPA (n; ) := dh(t K n ()) dh() where p i =mpol( i ) Q[x] (i {1 2 3}) is the monic minimal polynomial of i. We computed the length of the period of the expansion of ( 4 m 4 m 2 4 m 3 ) obtained by T K for m all 2 m 5000 and relative decimal heights given above; cf. Table D. We confirmed that ( 4 m 4 m 2 4 m 3 ) P K for all 2 m 5000( m/ Q).

9 A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM A conjecture Conjecture. Let K be a real cubic field or a real quartic field. Then: (1) X K = P K. (2) There exists an absolute constant c independent of K and X K such that rdh AJPA () c holds. (Probably we can take c =6.) (3) There are infinitely many X K (including ( )) such that the expansion of obtained by the Jacobi-Perron Algorithm modified by Podsypanin is not periodic. In Table A L(m 1 m 2 ) H(m 1 m 2 ) and R(m 1 m 2 ) are numbers defined by L(m 1 m 2 ) := the maximum value of the length of the shortest period of the expansion of ( 3 m 3 m 2 ) by our algorithm for m 1 m m 2 with 3 m/ Q H(m 1 m 2 ):= max dh m 1 m m 2 3 AJPA (n; 3 m 3 m 2 ) m/ Q0 n< R(m 1 m 2 ):= max rdh m 1 m m 2 3 AJPA (n; 3 m 3 m 2 ) m/ Q0 n< which are well defined by the periodicity. Table A range of m L(m 1 m 2 ) H(m 1 m 2 ) R(m 1 m 2 ) m 1 m m 2 2 m /2 201 m /2 401 m /2 601 m /5 801 m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m /3

10 2218 JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI In Table B U(m 1 m 2 ) V (m 1 m 2 ) are numbers defined by U(m 1 m 2 ):= max dh( T n ( ) m 1 n m 2 V (m 1 m 2 ):= min dh( T n ( ). m 1 n m 2 We define integral-valued functions U and V by U(x) :=U(1000n (n 1 +1)) if 1000n 1 <x 1000(n 1 +1) V (x) :=V (1000n (n 1 +1)) if 1000n 1 <x 1000(n 1 +1) for x n 1 Z 0. Table B range of n U(m 1 m 2 ) V (m 1 m 2 ) 0 n n n n n n n n n n n n n n n n n n n n

11 A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM Figure 1. The graph of U Figure 2. The graph of V

12 2220 JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI In Table C L(m 1 m 2 ) H(m 1 m 2 ) and R(m 1 m 2 ) are numbers defined by L(m 1 m 2 ) := the maximum value of the length of the shortest period of the expansion of ( τ m τ 2 m ) by our algorithm for m 1 m m 2 whereτ m is a maximal root of x 3 mx +1 H(m 1 m 2 ):= max dh AJPA(n; τ m τ m 1 m m 2 m ) 2 0 n< R(m 1 m 2 ):= max rdh AJPA(n; τ m τm ) 2 m 1 m m 2 0 n< which are well defined by the periodicity. In Table D L(m 1 m 2 ) H(m 1 m 2 ) and R(m 1 m 2 ) are numbers defined by L(m 1 m 2 ) := the maximum value of the length of the shortest period of the expansion of ( 4 m 4 m 2 4 m 3 ) by our algorithm for m 1 m m 2 with m/ Q H(m 1 m 2 ):= max m 1 m m 2 dh AJPA (n; 3 m 3 m 2 3 m 3 ) m/ Q0 n< R(m 1 m 2 ):= max m 1 m m 2 rdh AJPA (n; 3 m 3 m 2 3 m 3 ) m/ Q0 n< which are well defined by the periodicity. Table C range of m L(m 1 m 2 ) H(m 1 m 2 ) R(m 1 m 2 ) m 1 m m 2 3 m m /3 401 m /3 601 m /4 801 m m m m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m /2

13 A NEW MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHM 2221 Table D range of m L(m 1 m 2 ) H(m 1 m 2 ) R(m 1 m 2 ) m 1 m m 2 2 m /5 201 m m /6 601 m /7 801 m / m m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m / m /3 References 1. L. Bernstein The Jacobi-Perron algorithm Its theory and application. Lecture Notes in Mathematics Vol. 207 Springer-Verlag Berlin New York (1971). MR (44:2696) 2. L. Elsner H. Hasse Numerische Ergebnisse zum Jacobischen Kettenbruchalgorithmus in reinkubischen Zahlkörpern. (German) Math. Nachr. 34 (1967) MR (36:2589) 3. T. Fujita Sh. Ito M. Keane and M. Ohtsuki On almost everywhere exponential convergence of modified Jacobi-Perron algorithm: a corrected proof Ergodic theory an dynamical systems. 16 (1996) MR (98a:11105) 4. S. Ito J. Fujii H. Higashino and S. Yasutomi On simultaneous approximation to ( 2 )with 3 + k 1 = 0 J. Number Theory 1 (2003) MR (2004a:11063) 5. S. Ito and S. Yasutomi On simultaneous approximation to certain periodic points related to modified Jacobi-Perron algorithm to appear in Advanced Studies in Pure Mathematics. 6. GiNaC web site: 7. D.N. Lehmer On Jacobi s Extension of the Continued Fraction Algorithm Proc Natl Acad Sci U S A December; 4(12): C.D. Olds Continued fractions. Random House New York MR (26:3672) 9. O. Perron Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus Math. Ann. 64 (1907) MR

14 2222 JUN-ICHI TAMURA AND SHIN-ICHI YASUTOMI 10. E.V. Podsypanin A generalization of continued fraction algorithm that is related to the Viggo Brun algorithm (Russian) Studies in Number Theory (LOMI) 4 Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 67 (1977) MR (56:15545) 11. F. Schweiger Multidimensional continued fractions. Oxford Science Publications. Oxford University Press Oxford MR (2005i:11090) 12. J. Tamura A new approach to higher dimensional continued fractions preprint. 13. J. Tamura A class of transcendental numbers having explicit g-adic and Jacobi-Perron expansions of arbitrary dimension. Acta Arith. 71 (1995) no MR (96g:11083) Azamino Aoba-ku Yokohama Japan address: jtamura@tsuda.ac.jp General Education Suzuka National College of Technology Shiroko Suzuka Mie Japan address: yasutomi@genl.suzuka-ct.ac.jp