A class of transcendental numbers with explicit g-adic expansion and the Jacobi Perron algorithm

Transcription

1 ACTA ARITHMETICA LXI. (992) A class of transcendental numbers with explicit g-adic expansion and the Jacobi Perron algorithm by Jun-ichi Tamura (Tokyo). Introduction. In this paper we give transcendental numbers ϕ and ψ such that (i) both ϕ and ψ have explicit g-adic expansions and simultaneously (ii) the vector t (ϕ ψ) has an explicit expression in the Jacobi Perron algorithm (cf. Theorem ). Our results can be regarded as a higher-dimensional version of some of the results in [] [5] (see also [6] [8] [0] []). The numbers ϕ and ψ have some connection with algebraic numbers with minimal polynomials x 3 kx 2 lx satisfying (.) k l 0 k + l 2 (k l Z). In the special case k l our Theorems 3 have been shown in [5] by a different method using the theory of representation of numbers by Fibonacci numbers of third degree. 2. Notations and main results. We begin with some notations necessary to state our theorems. Let K be an alphabet i.e. a non-empty finite set of symbols. K (n) and K ω denote the set of all finite words w with length w n and the set of all ω-words w w 2... w i... (w i K). In particular K (0) {λ} where λ is the empty word. We put K : K (n) K + : K (n) K : K K ω. n0 n K is a free monoid generated by K with the operation of concatenation and with the unit λ. K is a complete metric space with metric d defined by d(u v) : (inf{n ; u n v n }) u v for u u u 2... u i... v v v 2... v j... K (u i v j K).

2 52 J. Tamura In what follows we set K : {a b c} and k l denote fixed integers satisfying (.). We define a monoid morphism σ Hom(K K ) by (2.) σ(a) a k b σ(b) a l c σ(c) a. G denotes the triple (K σ a). G is a DOL system (cf. [3] [4]). It is easily seen that (2.2) σ n+2 (a) (σ n+ (a)) k (σ n (a)) l σ n (a) n where σ n denotes the n-fold iteration of σ so that {σ n (a)} n0 is a Cauchy sequence in K. We write w : lim m σm (a) w w 2... w n... K ω (w n K) which will be referred to as the ω-word generated by G. We denote the word w by w(a b c) to specify the symbols a b c. For an integer g 2 let w(u u 2 u 3 ) stand for the infinite sequence of integers in {0... g } : L say obtained by replacing the symbols a b c appearing in w(a b c) with u u 2 u 3 L + respectively. Define the number 0.w(u u 2 u 3 ) by means of the g-adic expansion: 0.w(u u 2 u 3 ) : c n g n n where w(u u 2 u 3 ) c c 2... c n... L ω (c n L). We define the linear recurrence sequence {f n } n 2 by (2.3) Then it is easily seen that f n kf n + lf n 2 + f n 3 n N (f 2 f f 0 ) (l 2 l k + l ). (2.4) σ n (a) f n n N. We denote by A n B n C n (resp. A B C) the characteristic sets of the word σ n (a) w w 2... w m... w fn (resp. w w w 2... w m... K ω ) defined by (2.5) A A n : {m ; w m a m f n } B n : {m ; w m b m f n } C n : {m ; w m c m f n } A n B B n C C n. n n It is clear that A B and C give a partition of N. In this paper we consider the algorithm of Jacobi Perron (resp. of Parusnikov) by means of which a given vector t (ϕ ψ) R 2 (resp. t (ϕ ψ) K 2 where K C((z )) the field of formal Laurent series) can be developed n

3 in a higher-dimensional continued fraction [ ] a0 ; a a 2... a n... b 0 ; b b 2... b n... Transcendental numbers 53 where a i b i Z (resp. a i b i C[z]). Precise notations and definitions of these algorithms are given in Section 3. Now we can state our results: by (2.6) Theorem. Let g be an integer and let a n b n be integers defined l a n g f n 3 g fn 2 j n 3 j0 k b n g lf n 2+f n 3 g fn j n 2 j0 where f n is the sequence given by (2.3) with the conditions (.). Let ϕ(g) and ψ(g) be the numbers defined by the expression in the Jacobi Perron algorithm ϕ(g) 0; a3... a n... (2.7) :. ψ(g) 0; b 2 b 3... b n... The we have the following statements: (i) When g 2 ϕ(g) and ψ(g) are transcendental numbers and their g-adic expansions are given by (2.8) ϕ(g) 0.w(g 0 0) ψ(g) 0.w(g g 0). Moreover ϕ() and ψ() are algebraic numbers of the third degree where w is the ω-word generated by the DOL system G (K σ a). (ii) ϕ(g) and ψ(g) are linearly independent over Q for each g N. (iii) The expression (2.7) is admissible i.e. the right-hand side of (2.7) is derived from the Jacobi Perron algorithm. (iv) The n-th convergent of (2.7) is given by ( f n g m) t( m m A n g m m A n B n g m Theorem 2. Let A B and C be the characteristic sets of the word w K ω given by (2.5) and let η(z) ξ(z) be the functions defined by ). η(z) ϕ B (z)/ϕ A (z) ξ(z) (ϕ B (z) + ϕ C (z))/ϕ A (z) where ϕ D (z) denotes the function m D z m for a given subset D of N. Then we have the following statements:

4 54 J. Tamura (i) ϕ A (z) ϕ B (z) and ϕ C (z) are transcendental functions over C(z) with the unit circle z as their natural boundary. (ii) ϕ A (z) ϕ B (z) and ϕ C (z) are linearly independent over C(z). (iii) The admissible Jacobi Perron Parusnikov expansion for t (η(z) ξ(z)) is given by η(z) 0; a3 a 4... a n... (2.9) : ξ(z) 0; b 2 b 3 b 4... b n... where a n (n 3) and b n (n 2) are polynomials in z obtained by replacing g by z in (2.6). (iv) The n-th convergent of (2.9) is given by ( ) t( ) z m z m z m. m A n m B n m B n C n The transcendence result (i) in Theorem can be generalized: Theorem 3. Let g 2 be an integer and let L {0... g }. For any given u u 2 u 3 L + M denotes the matrix ( u i j ) i 3 0 j g where u i j indicates the number of j L appearing in the word u i. Suppose that rank M 2. Then the number defined by the g-adic expansion 0.w(u u 2 u 3 ) is transcendental. 3. Jacobi Perron algorithms. In this section we define two kinds of continued fraction expansions of higher dimension. We use the following notations. K C((z )) the field of formal Laurent series with complex coefficients. K is a metric space with the distance function d K (ϕ ψ) : ϕ ψ (ϕ ψ K) where is the usual non-archimedean norm defined by ϕ : e k for 0 ϕ mk c mz m K with c k 0 k Z and 0 : 0. [ϕ] : the polynomial part of ϕ K i.e. [ϕ] : 0 for ϕ < and [ϕ] : 0 mk c mz m for ϕ with ϕ mk c mz m. ϕ : ϕ [ϕ] the fractional part of ϕ K. We denote by [ϕ] and ϕ the vectors t ([ϕ] [ψ]) and t ( ϕ ψ ) respectively for ϕ t (ϕ ψ) K 2 where t (ϕ ψ) is the transposed vector of (ϕ ψ). T denotes the transformation defined by T ( t (ϕ ψ)) : t (/ψ ϕ/ψ) (0 ψ ϕ K). Furthermore we denote T ( t (ϕ ψ)) by / t (ϕ ψ). Following Parusnikov [2 9] we define the expansion of ϕ t (ϕ ψ) K 2 in the Jacobi Perron algorithm. We put ϕ 0 t (ϕ 0 ψ 0 ) : t ( ϕ ψ ) and ϕ n t (ϕ n ψ n ) : S n (ϕ 0 )

5 Transcendental numbers 55 if ϕ n 0 for all n 0 where S n denotes the n-fold iteration of S which is defined by S(ϕ 0 ) : T (ϕ 0 ) (S 0 the identity map on K). Then we define a n t (a n b n ) (C[z]) 2 by (3.) a 0 : [ϕ] a n : [T (S n (ϕ 0 ))] (n ). The vector π n (C(z)) 2 defined by (3.2) π n π n (ϕ) a 0 + a + a a n converges componentwise to ϕ as n by Parusnikov s theorem in [2] where the convergence is with respect to the metric d K. Thus we may write (3.3) ϕ a 0 + a + a a n + (3.2) and (3.3) will be also written as ϕ a0 ; a a 2... a n π n ψ b 0 ; b b 2... b n ϕ a0 ; a a 2... a n... :. ψ b 0 ; b b 2... b n... The right-hand side of (3.3) will be called the expression of ϕ K 2 in the Jacobi Perron Parusnikov algorithm (abbr. JPP algorithm). Apart from the algorithm given by (3.) we can consider the expression (3.3) for any given sequence of vectors a a 2... in K 2 provided that its nth convergent (3.2) is well-defined and converges to some element of K 2. The JPP expression (3.3) will be called admissible if it is derived from the algorithm (3.). Denote by A n and P n the 3 3 matrices given by A n : a n SL(3 C[z]) 0 b n P n : A A 2... A n n...

6 56 J. Tamura and P 0 : E the 3 3 unit matrix. Set P n : p n 2 p n p n q n 2 q n q n SL(3 C[z]). r n 2 r n r n Then we have ( ) ϕ (3.4) π n ψ ( a0 b 0 ) + ( ) pn /r n (C(z)) 2 q n /r n where {p n } n 2 {q n } n 2 {r n } n 2 are the sequences of polynomials satisfying the recurrence relations (3.5) p n b n p n + a n p n 2 + p n 3 n and the same recurrences with q n and r n in place of p n with the initial condition P 0 E. If we take the field R of real numbers for K and define [ϕ] to be the integral part of ϕ R we have an algorithm which is the simplest Jacobi Perron algorithm (abbr. JP algorithm). 4. Pisot numbers of third degree Lemma 4.. Let p(x) x 3 kx 2 lx be a polynomial satisfying the conditions (.). Then p(x) is irreducible over Q and has a real root α such that k < α < k + and β γ < where β γ are the algebraic conjugates of α i.e. α is a Pisot number. P r o o f. Since p(k) and p(k + ) k p(x) has a real root α with k < α < k +. Put ξ : (k k 2 + 3l)/3 so that (d/dx)p(ξ) 0 and 2/3 < ξ 0. We have < β γ < 0 if p(ξ) 0 otherwise β γ R β γ / α < since p( ) k + l 2 < 0 and p(0) < 0. In any case α β γ Z. Thus p(x) is irreducible. R e m a r k. It can be proved that the inequalities k l > 4k + 3 (k 5) imply β γ R and 3k l (k 3) implies β γ R. Lemma 4.2. Let λ /α and k l/α + /α 2 where α > is the root of p(x) given in Lemma 4.. Then the following expression is admissible in the JP algorithm: λ 0; l l l... (4.) :. κ 0; k k k... Furthermore writing π n t (λ κ) t (p n /r n q n /r n ) as in Section 3 we have (4.2) λ p n /r n O(r δ n ) κ q n /r n O(r δ n ) where δ : log β / log α (> 0) and β γ are the algebraic conjugates of α with β γ.

7 P r o o f. We have Transcendental numbers 57 T ( t (λ κ)) t (l + /α α) t (l k) + t (λ κ) with 0 < λ < and 0 < κ < which leads to the admissible expression (4.). Noting that the characteristic polynomial of the recurrence sequences p n q n r n is p(x) we can write with positive numbers µ ν. Since we have Similarly we get p n µα n + O( β n ) q n να n + O( β n ) r n p n+ µα n + O( β n ) λr n p n κ(p n r n p n r n ) + λ(p n 2 r n p n r n 2 ) r n + κr n + λr n 2 (4.2) follows from these inequalities. λr n p n O( β n ). κr n p n O( β n ). 5. DOL system G and related polynomials. We define the polynomials p n p n (z) q n q n (z) r n r n (z) by (5.) p n : z m q n : z m r n : z m m A n m B n m C n where A n B n C n are the characteristic sets of σ n (a) defined by (2.5). For brevity we put (h) m : m i0 Then from (2.2) and (2.4) we have Lemma 5.. h i (h) 0 : 0 (h C(z)). (5.2) p n c n p n + b n p n 2 + ã n p n 3 n 4 and the same recurrences hold with q n and r n in place of p n with the initial conditions (5.3) p p 2 p 3 z z (z) k z (z) k (z k+ ) k + z k(k+)+ (z) l q q 2 q 3 0 z k+ z k+ (z k+ ) k r r 2 r z k2 +k+l+

8 58 J. Tamura where (5.4) ã n z kf n +lf n 2 ( z f n f n 3 ) bn z kf n (z f n 2 ) l c n (z f n ) k and f n is the sequence defined by (2.3). (5.5) We put p n (z) z fn p n (z ) q n (z) z fn q n (z ) r n (z) z fn r n (z ). Then p n q n and r n are polynomials in z. It follows from (5.2) that p n (z) z f n f n 3ã n (z )p n 3 (z) + z f n f n 2 bn (z )p n 2 (z) + z f n f n c n (z )p n (z) which together with (5.4) yields the recurrence relations (5.6) p n b n p n + a n p n 2 + p n 3 n 4 and the same recurrences with q n and r n in place of p n where (5.7) a n z f n 3 (z f n 2 ) l b n z lf n 2+f n 3 (z f n ) k. In view of (5.3) we have the initial conditions (5.8) p p 2 p 3 q q 2 q 3 z (z) k z (z) k z l+ (z k+ ) k + z (z) l 0 z l+ (z k+ ) k r r 2 r Now we set P n : p n 2 p n p n q n 2 q n q n : 0 0 a 0 r n 2 r n r n b a 2 (5.9) 0 0 P 0 : a 0. a a 2 b a 2 Then the relations (5.6) for p n q n and r n hold for all n by (5.7) (5.8) and (2.3). Therefore we get P n A A 2... A n (n N) P 0.

9 where Transcendental numbers 59 A n : a n. 0 b n We define the sequence of rational functions p n p n (z) q n q n (z) and r n r n (z) by (5.0) P n : P n P n : p n 2 p n p n q n 2 q n q n n 0. r n 2 r n r n By the definitions (5.9) and (5.0) we have P n A A 2... A n (n N) P 0 E where E is the 3 3 unit matrix. Therefore t (p n (z)/r n (z) q n (z)/r n (z)) is the nth convergent of the JPP expression ϕ(z) 0; a a 2... a n... (5.) ψ(z) 0; b b 2... b n... (see (3.4) (3.5)). It follows from (5.0) (5.5) and (5.) that and p n (z)/r n (z) p n (z )/(b (z) p n (z ) + a 2 (z) q n (z ) + r n (z )) q n (z)/r n (z) (a (z) p n (z ) + q n (z ))/(b (z) p n (z ) + a 2 (z) q n (z ) + r n (z )) lim p n(z ) z m m A lim r n(z ) z m. m C lim q n(z ) z m m B Since the above series converge in the region z > the right-hand side expression in (5.) converges in z > and hence ϕ(z) and ψ(z) are holomorphic functions on z > given by m A ϕ(z) z m b (z) m A z m + a 2 (z) m B z m + m C z m (5.2) ψ(z) a (z) m A z m + m B z m b (z) m A z m + a 2 (z) m B z m + m C z m. We note that the partial denominators a n (n 3) b n (n 2) in the JPP expression (5.) are polynomials in z. But a a 2 b may not be

10 60 J. Tamura polynomials for certain k l. However it follows from (5.2) that ( ) T pn (z)/r n (z) q n (z)/r n (z) ( ) ( a (z) q + n (z )/ p n (z ) b (z) (a 2 (z) q n (z ) + r n (z ))/ p n (z ) ( ( ) ( )) T T pn (z)/r n (z) a (z) q n (z)/r n (z) b (z) ( a2 (z) 0 ) + n 3. Thus we have ( rn (z )/ q n (z ) [ ] ) 0; a3 a 4... a n (5.3) p n (z )/ q n (z ) b 2 ; b 3 b 4... b n ) ( ) rn (z )/ q n (z ) p n (z )/ q n (z ) since t (p n (z)/r n (z) q n (z)/r n (z)) is the nth convergent of (5.). On the other hand ( ) ( T ϕn (z) m A n B n z m / m A n z m ) ψ n (z) m A n B n C n z m / m A n z m ( ) ( m B + n z m / m A n z m ) m B n C n z m / m A n z m n where ϕ n (z) and ψ n (z) are the rational functions given by ϕ n (z) : / z m z m m A n m A n B n C n (5.4) ψ n (z) : / z m z m. m A n B n m A n B n C n Hence we have ( ( ) (5.5) T T ϕn (z) ψ n (z) ( )) ( 0 ) ( m C + n z m / m B n z m ) m A n z m / m B n z m n 2. By (5.5) (5.) and (5.3) we get ( ( ) ( )) [ ] T T ϕn (z) ; a3... a n n 3 ψ n (z) b 2 ; b 3... b n that is ϕn (z) 0; a3... a n (5.6) ψ n (z) 0; b 2 b 3... b n which holds for all n.

11 6. Proof of the theorems Transcendental numbers 6 P r o o f o f T h e o r e m. We define the functions ϕ(z) and ψ(z) by ϕ(z) 0; a3... a n... (6.) ψ(z) 0; b 2 b 3... b n... where a n (n 3) and b n (n 2) are the polynomials in z given by (5.7). Since (5.6) is the nth convergent of (6.) we have by (5.4) lim ϕ n(z) (z ) z m ϕ(z) (6.2) lim ψ n(z) (z ) m A m A B z m ψ(z) which are holomorphic functions on z >. Taking z g (2 g Z) in (6.2) we get (2.8). Furthermore it follows from (5.6) (5.7) (6.) and Lemma 4.2 that [ ] ϕ() 0; l l l... 0; + λ (6.3) ψ() 0; k k k k... 0; + κ where λ and κ are the numbers given in Lemma 4.2. Hence we have by simple calculation (6.4) ϕ() α 2 /(α 2 + α + ) ψ() (α 2 + α)/(α 2 + α + ) which are irrationals of third degree where α is the number given in Lemma 4.. Thus we obtain the statement (i) of Theorem except for the transcendence. (iv) is clear from (5.4) and (5.6). For the rest of the proof we prepare some lemmas. In view of (5.4) and (6.4) we get the following Lemma 6.. Let A n B n C n be the characteristic sets (2.5) of the word σ n (a) and let α be the root of p(x) as in Lemma 4.. Then lim A n /f n α 2 /(α 2 + α + ) lim B n /f n α/(α 2 + α + ) lim C n /f n /(α 2 + α + ) where D denotes the number of elements of a finite set D. These numbers are irrationals of third degree. We define the degree ϱ G of self-similarity of the DOL system G (K σ a) by ϱ G : lim sup d( lim m σm (a) σ n (a) ) σ n (a)

12 62 J. Tamura where σ n (a) stands for the infinite periodic word σ n (a)σ n (a)... K ω and d is the distance function of the metric space K described in Section 2. In the following we shall show that the quantity ϱ G is connected with the irrationality measure of rational approximation for the numbers ϕ(g) and ψ(g) of Theorem. Here we say that µ is an irrationality measure of a number θ R when θ p/q < q µ+ε for infinitely many coprime integers p and q for any fixed ε > 0. Lemma 6.2. Let G (K σ a) be the DOL system given in Section 2 and let α be the number given by Lemma 4.. Then k + /α (k 2 l ) ϱ G k + /α 2 (k 2 l 0) 2 + /α 3 (k l ). P r o o f. We put u n σ n (a) for brevity. Then it follows from (2.2) that w : lim m σm (a) u n+... u k nu l n u n 2... n 3. Since u n and u n 2 are prefixes of u n we have { k + fn /f ϱ G n (l > 0) k + f n 2 /f n (l 0). We get this lemma for k 2 by taking n. We write u v (u v K ) when v is a prefix of u. If k l we obtain w u n+2 u n+ u n u n u n u n u n 2 u n 2 u n 3 u n u n u n 2 u n 3 u n 4 u n 5 u n 3 u n u n u n 4 u n 5 u n 6 u n u n u n 3 for n 7 which leads to ϱ G 2 + /α 3. R e m a r k. By a more precise argument we obtain w u n u n u n 3 u n 6 u n 9... u n 3k n 3k + 4 when k l. Hence we can prove ϱ G 2 + /(α 3 ) when k l. Lemma 6.3. The irrationality measures of the numbers ϕ(g) and ψ(g) defined by (2.7) with 2 g Z are greater than 2. P r o o f. From (5.4) we have the following g-adic expansions of ϕ(g)

13 and ψ(g): (6.5) ϕ n (g) Transcendental numbers 63 ( (g )g f n m A n g m 0.u n(g 0 0) ( ψ n (g) (g )g f n m A n B n g m 0.u n(g g 0) )/ (g f n ) )/ (g f n ) where u n u n(a b c) is the infinite periodic word σ n (a)σ n (a)... K ω. Hence we deduce from (6.2) that (6.6) ϕ(g) ϕ n (g) < g (ϱ G ε)f n ψ(g) ψ n (g) < g (ϱ G ε)f n for all sufficiently large n for any fixed ε > 0. Let s n (g) > 0 and t n (g) > 0 be the denominators of ϕ n (g) and ψ n (g) respectively in the expressions of their irreducible fractions. Then it follows from (6.5) that s n (g) and t n (g) are divisors of g f n. Therefore ϕ(g) ϕ n (g) < s n (g) ϱ G+ε ψ(g) ψ n (g) < t n (g) ϱ G+ε for infinitely many n for any fixed ε > 0. Both {ϕ n (g)} n and {ψ n (g)} n are infinite sets since ϕ(g) ψ(g) Q which follows from Lemma 6.. In view of Lemma 6.2 the proof of the lemma is complete. Lemma 6.3 together with Roth s theorem implies the transcendence of ϕ(g) and ψ(g); this completes the proof of (i) of Theorem. Next we prove the statement (ii) of Theorem. It follows from Lemma 4.2 (6.3) and (6.6) that for any fixed g N r n (g)ϕ(g) p n (g) o() r n (g)ψ(g) q n (g) o() where p n q n r n are integers given by (5.0). Suppose that ϕ(g) and ψ(g) are linearly dependent over Q. Then x(p n /r n + o()/r n ) + y(q n /r n + o()/r n ) + z 0 and hence xp n + yq n + zr n 0 for all sufficiently large n for some fixed integers x y z with (x y z) (0 0 0). This contradicts the fact that 0 det P n ( ). It remains to prove (iii). Clearly (iii) holds for g. We put ϕ (n) (g) 0; an+ a n+2... ψ (n) g 2 n 0 (g) 0; b n+ b n+2... where a n (n 3) b n (n 2) are the integers given in Theorem with a a 2 b. Then (6.2) implies p < ϕ (0) (g) ϕ(g) < 0 < ψ (0) (g) ψ(g) < n. In general both ϕ (n) (g) and ψ (n) (g) (n Z + ) are different

14 64 J. Tamura from 0 and otherwise the expansion of t (ϕ(g) ψ(g)) in the JP algorithm is terminating which contradicts the linear independence of ϕ(g) and ψ(g) over Q. Hence and therefore 0 < ϕ (n) (g) /(b n+ + ψ (n+) (g)) /b n+ < n 0 < ψ (n) (g) (a n+ + ϕ (n+) (g))/(b n+ + ψ (n+) (g)) < (a n+ + )/b n+ ( l )/( k ) + g f n 2 g f n m g lf n g f n 2 g f nm m0 so that 0 < ψ (n) (g) < if l 0 and /( k 0 < ψ (n) (g) g f n 2 if l 0 k >. This implies (iii). m0 g f nm ) < m0 P r o o f o f T h e o r e m 2. Although the expression (6.) is admissible in the JP algorithm when z g for any integer g N the expression (6.) is not admissible in the JPP algorithm. Applying T and subtracting t ( ) from both sides of (6.) we get ψ(z)/ϕ(z) 0; a3.... /ϕ(z) 0; b 2 b 3... Recalling that A B C ( N) is a disjoint union we get the expression (2.9) of Theorem 2. We also get (iv) of Theorem 2 in view of (5.4) and (5.6). Next we show the admissibility of (2.9). Let η n (m) (z) ξ n (m) (z) be the functions given by ( (m) ) [ ] η n (z) 0; am a m+... a m+n (z) 0; b m b m+... b m+n ξ (m) n with m 2 n and a 2 where a n (n 3) and b n (n 2) are the polynomials in z in Theorem 2. Then we have (6.7) η (m) n (z) exp( deg b m ) e m 2 n ξ (m) n (z) exp(deg a m deg b m ) e m 2 n by using (3.4) (5.7) and the following lemma which is easily proved by induction on n. Lemma 6.4 (Parusnikov [2]). Let p n q n r n (n 2) be the polynomials given by (3.5) with a n b n C[z] (n ) such that 0 deg a n < deg b n.

15 Then for all n. From (6.7) we have deg p n Transcendental numbers 65 n deg b m m2 deg q n deg a + deg r n n deg b m m2 n deg b m m lim η(m) n < lim ξ(m) n < m 2 where the convergence follows from that of (6.). It is clear that the above inequalities hold for m. Thus we get the admissibility of the expression (2.9) in the JPP algorithm. We now prove (ii). The linear independence of the functions η(z) and ξ(z) can be proved by the same arguments as for the numbers ϕ(g) and ψ(g) by using the inequalities corresponding to (6.7) in the case of functions in K C((z )) with metric d K (cf. Parusnikov s theorem [2]). From this (ii) follows. In view of Lemma 6. ϕ A (z) ϕ B (z) and ϕ C (z) given in Theorem 2 are not rational functions which together with the following lemma implies the transcendence result (i) in Theorem 2. The proof of Theorem 2 is complete. Lemma 6.5 (Carlson Pólya). Let f(z) be a function given by a power series n0 c nz n (c n Z) with lim sup c n /n. Then the unit circle z is the natural boundary of f(z) except when f(z) is a rational function. P r o o f o f T h e o r e m 3. We need the following Lemma 6.6. Let L {0... g } be the alphabet of g symbols with g 2 and let u u 2 u 3 L +. Define the infinite word v v v 2... v n... L ω (v n L) by v w(u u 2 u 3 ) where w w(a b c) is the ω-word generated by the DOL system G given in Section 2. Let M denote the matrix ( u i j ) i 3 0 j g where u i j denotes the number defined in Theorem 3. Suppose that rank M 2. Then v is not ultimately periodic. P r o o f. Put N N(n) 3 i A (i) n u i

16 66 J. Tamura where A () n A n A (2) n B n A (3) n C n which are the characteristic sets of σ n (a) defined by (2.5) and A (i) n denotes the number of elements of A (i) n. Then we have Hence we get v v 2... v N(n) j lim N(n) v v 2... v N j 3 i A (i) n u i j. 3 / 3 α 3 i u i j α 3 i u i ζ j say i in view of Lemma 6.. Since α is an irrational number of third degree ζ j is rational if and only if the vectors ( u u 2 u 3 ) and ( u j u 2 j u 3 j ) are linearly dependent over Q. Hence ζ j is irrational for some j (0 j g ) since g j0 u i j u i and rank M 2. Therefore v is not ultimately periodic. We can prove Theorem 3 by applying Lemmas 6.2 and 6.6 together with Roth s theorem considering the rational approximation of 0.w(u u 2 u 3 ) 0.v v 2... v m... by the rational numbers with periodic digits 0. v v 2... v N(n) (n ) in their g-adic expansion. As we have already seen the monoid morphism σ defined by (2.) has some connection with the Pisot number with minimal polynomial x 3 kx 2 lx satisfying (.). We can state theorems similar to Theorems 3 relative to the JP and JPP algorithms of dimension s and Pisot numbers with minimal polynomial x s k s x s k s x s... k x under the conditions k k 2... k s (k i Z) for any given s. We will give such results in forthcoming papers. i References [] W. W. Adams and J. L. Davison A remarkable class of continued fractions Proc. Amer. Math. Soc. 65 (977) [2] P. E. Böhmer Über die Transzendenz gewisser dyadischer Brüche Math. Ann. 96 (927) ; Berichtigung ibid [3] P. Bundschuh Über eine Klasse reeller transzendenter Zahlen mit explizit angebbarer g-adischer und Kettenbruch-Entwicklung J. Reine Angew. Math. 38 (980) 0 9. [4] L. V. D a n i l o v Some classes of transcendental numbers Mat. Zametki 2 (2) (972) (in Russian); English transl.: Math. Notes 2 (972) [5] J. L. Davison A series and its associated continued fraction Proc. Amer. Math. Soc. 63 (977) [6] J. H. Loxton and A. J. van der Poorten Arithmetic properties of certain functions in several variables III Bull. Austral. Math. Soc. 6 (977) 5 47.

17 Transcendental numbers 67 [7] K. M a h l e r Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen Math. Ann. 0 (929) [8] D. Masser A vanishing theorem for power series Invent. Math. 67 (982) [9] E. M. Nikishin and V. N. Sorokin Rational Approximations and Orthogonality Nauka Moscow (in Russian). [0] K. Nishioka Evertse theorem in algebraic independence Arch. Math. (Basel) 53 (989) [] K. Nishioka I. Shiokawa and J. Tamura Arithmetical properties of certain power series J. Number Theory to appear. [2] V. I. Parusnikov The Jacobi Perron algorithm and simultaneous approximation of functions Mat. Sb. 4 (56) (2) (98) (in Russian). [3] A. Salomaa Jewels of Formal Language Theory Pitman 98. [4] Computation and Automata Cambridge Univ. Press 985. [5] J. T a m u r a Transcendental numbers having explicit g-adic and Jacobi Perron expansions in: Séminaire de Théorie des Nombres de Bordeaux to appear. FACULTY OF GENERAL EDUCATION INTERNATIONAL JUNIOR COLLEGE EKODA 4-5- NAKANO-KU TOKYO 65 JAPAN Received on and in revised form on (204)