Transcendence with Rosen continued fractions
aa r X i v : . [ m a t h . N T ] J u l TRANSCENDENCE WITH ROSEN CONTINUEDFRACTIONS
YANN BUGEAUD, PASCAL HUBERT, AND THOMAS A. SCHMIDT
Abstract.
We give the first transcendence results for the Rosen con-tinued fractions. Introduced over half a century ago, these fractionsexpand real numbers in terms of certain algebraic numbers. Introduction
In 1954, D. Rosen defined an infinite family of continued fraction algo-rithms [20]. Introduced to aid in the study of certain Fuchsian groups, thesecontinued fractions were applied some thirty years later by J. Lehner [14]in the study of Diophantine approximation by orbits of these groups.The Rosen continued fractions and variants have been of recent inter-est, leading to results especially about their dynamical and arithmeticalproperties, see [9], [18], [11]; as well on their applications to the studyof geodesics on related hyperbolic surfaces, see [23], [8], [17]; and to Te-ichm¨uller geodesics arising from (Veech) translation surfaces, see [26], [5],[25] and [6]. Several basic questions remain open, including that of arith-metically characterizing the real numbers having a finite Rosen continuedfraction expansion, see [15], [13] and [6]. Background on Rosen continuedfractions is given in the next section.The first transcendence criteria for regular continued fractions were provedby E. Maillet, H. Davenport and K. F. Roth, A. Baker, and recently im-proved by B. Adamczewski and Y. Bugeaud, see [1, 2] and the references
Date : 24 June 2010.2000
Mathematics Subject Classification.
Key words and phrases.
Rosen continued fractions, Liouville inequality, Hecke groups,transcendence.The second named author is partially supported by project blanc ANR: ANR-06-BLAN-0038. The third author thanks FRUMAN, Marseille and the Universit´e P.C´ezanne. given there. In particular, Theorem 4.1 of [2] asserts that, if ξ is an alge-braic irrational number with sequence of convergents ( p n /q n ) n ≥ , then thesequence ( q n ) n ≥ cannot increase too rapidly. It is natural to ask whethersimilar transcendence results can be proven using Rosen continued fractions.We give the first such results. Theorem 1.1.
Fix λ = 2 cos π/m for an integer m > , and denote thefield extension degree [ Q ( λ ) : Q ] by D . If a real number ξ / ∈ Q ( λ ) hasan infinite expansion in Rosen continued fraction over Q ( λ ) of convergents p n /q n satisfying lim sup n →∞ log log q n n > log(2 D − , then ξ is transcendental. For stating our second result, we associate to the Rosen continued fractionexpansion[ ε ( x ) : r ( x ) , ε ( x ) : r ( x ) , . . . , ε n ( x ) : r n ( x ) , . . . ] := ε r λ + ε r λ + · · · of a real number x the sequence of pairs of integers ( ε i , r i ) i ≥ , which we callthe partial quotients , and thus consider A = {± } × N as the alphabet ofthe Rosen continued fraction expansions.As usual, we denote the length of a finite word U = u · · · u k as | U | = k .For any positive integer s , we write U s for the word U . . . U ( s times repeatedconcatenation of the word U ). More generally, for any positive real number s , we denote by U s the word U ⌊ s ⌋ U ′ , where U ′ is the prefix of U of length ⌈ ( s − ⌊ s ⌋ ) | U |⌉ .Just as Adamczewski and Bugeaud [1] showed for regular continued frac-tion expansions, a real number whose Rosen continued fraction expansionis appropriately “stammering” must be transcendental. Theorem 1.2.
Fix λ = 2 cos π/m for an integer m > , and denote thefield extension degree [ Q ( λ ) : Q ] by D . Let ξ be an infinite Rosen continuedfraction with convergents ( p n /q n ) n ≥ such that B := lim sup n q /nn < + ∞ . RANSCENDENCE WITH ROSEN CONTINUED FRACTIONS 3
Write b := lim inf n q /nn . Assume that there are two infinite sequences ( U n ) n ≥ and ( V n ) n ≥ of finitewords over the alphabet A and an infinite sequence ( w n ) n ≥ of real numbersgreater than such that, for n ≥ , the word U n V w n n is a prefix of the infiniteword composed of the partial quotients of ξ . If (1) lim sup n → + ∞ | U n | + w n | V n | | U n | + | V n | > D · log B log b , then ξ is either (at most) quadratic over Q ( λ ) or is transcendental. Lemma 2.1 gives that log b is positive.The key to our proofs is that both numerator and denominator of aRosen convergent dominate their respective conjugates in an appropriatefashion, see Lemma 3.1. From this one can bound the height of a Rosenconvergent in terms of its denominator, see Lemma 3.2. Then, exactly asin the case of regular continued fractions, we apply tools from Diophantineapproximation, namely an extension to number fields of the Roth theorem,for the proof of Theorem 1.1, and the Schmidt Subspace Theorem for theproof of Theorem 1.2.Both theorems are weaker than their analogues for regular continued frac-tions, since we have to work in a number field of degree D rather than in thefield Q . However, for m = 4 and m = 6, that is, for λ = √ λ = √ p n /q n , exactly oneof p n , q n is in Z , the other being in λ Z , see Remark 2 below.2. Background
Rosen fractions.
We set λ = λ m = 2 cos πm and I m = [ − λ/ , λ/ m ≥
3. For a fixed integer m ≥
3, the Rosen continued fraction map is
YANN BUGEAUD, PASCAL HUBERT, AND THOMAS A. SCHMIDT defined by T ( x ) = (cid:12)(cid:12) x (cid:12)(cid:12) − λ ⌊ (cid:12)(cid:12) λx (cid:12)(cid:12) + ⌋ x = 0;0 x = 0for x ∈ I m ; here and below, we omit the index “ m ” whenever it is clear fromcontext. For n ≥
1, we define ε n ( x ) = ε ( T n − x ) and r n ( x ) = r ( T n − x )with ε ( y ) = sgn( y ) and r ( y ) = (cid:22) (cid:12)(cid:12)(cid:12)(cid:12) λy (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:23) . Then, as Rosen showed in [20], the Rosen continued fraction expansion of x is given by[ ε ( x ) : r ( x ) , ε ( x ) : r ( x ) , . . . , ε n ( x ) : r n ( x ) , . . . ] := ε r λ + ε r λ + · · · . As usual we define the convergents p n /q n of x ∈ I m by (cid:18) p − p q − q (cid:19) = (cid:18) (cid:19) and (cid:18) p n − p n q n − q n (cid:19) = (cid:18) ε λr (cid:19) (cid:18) ε λr (cid:19) · · · (cid:18) ε n λr n (cid:19) for n ≥
1. From this definition it is immediate that | p n − q n − q n − p n | = 1,and that the well-known recurrence relations p − = 1; p = 0; p n = λr n p n − + ε n p n − , n ≥ q − = 0; q = 1; q n = λr n q n − + ε n q n − , n ≥ , hold. It also follows that(2) (cid:18) p n − q n − p n q n (cid:19) = (cid:18) ε n λr n (cid:19) (cid:18) ε n − λr n − (cid:19) · · · (cid:18) ε λr (cid:19) , giving(3) p n q n = [ ε : r , ε : r , . . . , ε n : r n ] RANSCENDENCE WITH ROSEN CONTINUED FRACTIONS 5 and(4) q n − q n = [ 1 : r n , ε n : r n − , . . . , ε : r ] . We define(5) M n = (cid:18) p n − p n q n − q n (cid:19) , and find that x = M n · T n ( x ) , where · denotes the usual fractional linear operation, namely x = p n − T n ( x ) + p n q n − T n ( x ) + q n . Approximation with Rosen fractions.
We briefly discuss the con-vergence of the “convergents” to x . One can rephrase some of Rosen’soriginal arguments in terms of the (standard number theoretic) natural ex-tension map T ( x, y ) = ( T ( x ) , rλ + εy ) where r = r ( x ) and ε = ε ( x ) . The“mirror formula” Equation (4) shows that T n ( x, y ) = ( T n ( x ) , q n − q n ) . Ex-tending earlier work of H. Nakada, it is shown in [9] that T ( x, y ) has planardomain with y -coordinates between 0 and R = R ( λ ) , where R = 1 if the in-dex m is even and, otherwise, R is the positive root of R +(2 − λ ) R − > R > λ/ q n ) n ≥ is strictly increasing. But, as Rosen mentions, if x hasinfinite expansion, then either ε n = 1 or r n > q n ≥ n and that the limit as n tends toinfinity of q n is infinite.One easily adapts Rosen’s arguments so as to find the following. Lemma 2.1.
For every x ∈ I m of infinite expansion, we have lim inf n q /nn > . Proof.
We know that the sequence ( q n ) n ≥ increases and that, if either ε n = 1 or r n >
1, then q n > λq n − . Furthermore, there are no more than h consecutive indices i with ( ε i , r i ) = ( − , h = m/ , ( m − / m (see [20] or [9]). Consequently, for any n ,there is some i = 1 , . . . , h + 1 such that q n + i > λq n + i − , giving q n + h +1 ≥ q n + i > λq n + i − ≥ λq n . YANN BUGEAUD, PASCAL HUBERT, AND THOMAS A. SCHMIDT As q ≥ λ , letting s ( n ) = 1 + (cid:22) n − h + 1 (cid:23) , we have q n ≥ λ s ( n ) . Since λ > (cid:3) Remark 1.
In fact, H. Nakada [19] shows that for almost all such x ,lim n →∞ n log q n exists, being equal to one half of the entropy of T . He alsoshows that the entropy equals C · ( m − π / (2 m ), where C = 1 / log(1 + R )when m is odd, and equals 1 / log[(1 + cos π/m ) / sin π/m ] when m is even.This C is the normalizing constant to give a probability measure on thedomain of the planar natural extension T , see [9].Rosen also gave bounds on | x − p n /q n | . Using Equation (5) (as in Nakada[19]) (cid:12)(cid:12)(cid:12)(cid:12) x − p n q n (cid:12)(cid:12)(cid:12)(cid:12) = 1 q n | q n +1 q n + T n +1 x | , one has the easy lower bound 1 q n ( q n +1 + q n ) < (cid:12)(cid:12)(cid:12)(cid:12) x − p n q n (cid:12)(cid:12)(cid:12)(cid:12) . One also finds in this manner(6) (cid:12)(cid:12)(cid:12)(cid:12) x − p n q n (cid:12)(cid:12)(cid:12)(cid:12) < c q n q n +1 , with c = c ( λ ) = 2 / (2 − Rλ ) . Thus convergence does hold.Theorems 4.4 and 4.5 (depending on parity of m ) of [9] give c such that | x − p n /q n | < c /q n , with the upper bound c ≤ / ⌈ m/ ⌉ .2.3. Traces in Hecke groups.
Rosen introduced his continued fractionsto study the Hecke groups. The Hecke (triangle Fuchsian) group G m with m ∈ { , , , . . . } is the group generated by (cid:18) λ m (cid:19) and (cid:18) −
11 0 (cid:19) , with λ m as above. The Rosen expansion of a real number terminates at afinite term if and only if x is a parabolic fixed point of G m , see [20]. Thesepoints are clearly contained in Q ( λ m ) but in general there are elements ofthis field that have infinite Rosen expansion, see [15], [13] and [6]. RANSCENDENCE WITH ROSEN CONTINUED FRACTIONS 7
Remark 2.
The values of finite Rosen expansion form the set G q · ∞ , whichis in fact a subset of λ Q ( λ ) ∪ {∞} . To see this, one uses induction on wordlength in the generators displayed above — an ordered pair ( a, c ) giving acolumn of any element of G q must be such that exactly one element of thepair is in Z [ λ ] , and the other is in λ Z [ λ ] . Note that this also appliesto convergents p n /q n : exactly one of p n , q n is in Z [ λ ], the other being in λ Z [ λ ] .When q = 3 , we have G = PSL(2 , Z ) . In general each G m is isomorphicto the free product of a cyclic group of order two and a cyclic group of order m . Recall that a Fuchsian triangle group is generated by even words in thereflections about the sides of some hyperbolic triangle. Thus any Fuchsiantriangle group is of index two in the group generated by these reflections;for each G m , we denote this larger group by ∆ m .Since λ m is the sum of the root of unity ζ m := exp 2 πi/ (2 m ) with itscomplex conjugate, Q ( λ m ) is a number field of degree d := φ (2 m ) / φ denotes the Euler totient function.The following key phenomenon property of Hecke groups can be shownin various manners. The result holds for a larger class of groups, fromCorollary 5 of [24], due to [10] (extending the arguments from G m to ∆ m isstraightforward). Independent of this earlier work, Bogomolny-Schmit [8]gave a clever proof of the result specifically for ∆ m . See the next remarkfor another perspective. Theorem 2.1.
Fix m as above, and let ∆ m be the full reflection group inwhich G m has index two. Then for any M ∈ ∆ m whose trace is of absolutevalue greater than , we have | tr ( M ) | ≥ | σ ( tr ( M ) ) | , where σ is any field embedding of Q ( λ m ) . Remark 3.
This result can be proven “geometrically”. Up to conjugacy,each of the Hecke groups appears as the Veech group of some translationsurface, see [26]; the elements whose trace is of absolute value at least 2are the “derivatives” of the affine pseudo-Anosov diffeomorphisms of the
YANN BUGEAUD, PASCAL HUBERT, AND THOMAS A. SCHMIDT surface. The dilatation of a pseudo-Anosov φ is the dominant eigenvalue λ of the action of φ on the integral homology of the underlying surface. (Theother eigenvalues are hence conjugates of λ .) The corresponding element ofthe Veech group has trace of absolute value λ + λ − from which it followsthat this trace dominates its conjugates.2.4. Approximation by algebraic numbers.
The following result wasannounced by Roth [22] and proven by LeVeque, see Chapter 4 of [16]. (Theversion below is Theorem 2.5 of [7].) Recall that given an algebraic number α , its naive height , denoted by H ( α ) , is the largest absolute value of thecoefficients of its minimal polynomial over Z . Theorem 2.2. (LeVeque) Let K be a number field, and ξ a real algebraicnumber not in K . Then, for any ǫ > , there exists a positive constant c ( ξ, K, ǫ ) such that | ξ − α | > c ( ξ, K, ǫ ) H ( α ) ǫ holds for every α in K . The logarithmic Weil height of α lying in a number field K of degree D over Q is h ( α ) = D P ν log + max ν ∈ M K {|| α || ν } , where log + t equals 0 if t ≤ M K denotes the places (finite and infinite “primes”) of the field, and || · || ν is the ν -absolute value. This definition is independent of the field K containing α . The two heights are related by(7) log H ( α ) ≤ deg( α ) h ( α ) + log 2 , for any non-zero algebraic number α , see Lemma 3.11 from [27].We recall a consequence of the W. Schmidt Subspace Theorem. Theorem 2.3.
Let d be a positive integer and ξ be a real algebraic numberof degree greater than d . Then, for every positive ε , there exist only finitelymany algebraic numbers α of degree at most d such that | ξ − α | < H ( α ) − d − − ε . RANSCENDENCE WITH ROSEN CONTINUED FRACTIONS 9
Note that the Roth theorem is exactly the case d = 1 of Theorem 2.3.In the proof of Theorem 1.2, we could apply Theorem 2.3, but the al-gebraic numbers α which we use to approximate ξ are of degree at most 2over a fixed number field. In this situation, the next theorem, kindly com-municated to us by Evertse [12], yields a stronger result than the previousone. Theorem 2.4. (Evertse) Let K be a real algebraic number field of degree d .Let t be a positive integer and ξ be a real algebraic number of degree greaterthan t over K . Then, for every positive ε , there exist only finitely manyalgebraic numbers α of degree t over K and δ over Q such that | ξ − α | < H ( α ) − dt ( t +1+ ε ) /δ . Note that Theorem 2.4 extends Theorem 2.2.2.5.
Sturmian sequences: towards an application of Theorem 1.2.
To give an explicit family of Rosen expansions satisfying the hypotheses ofTheorem 1.2, we recall a result of [3] on Sturmian sequences.Let a and b be letters in some alphabet. The complexity function of asequence u = u u · · · with values in { a, b } is given by letting p ( n, u ) be thenumber of distinct words of length n that occur in u . A sequence u is called Sturmian if its complexity satisfies p ( n, u ) = n + 1 for all n . As Arnoux[4] writes, one can obtain any such sequence by taking a ray with irrationalslope in the real plane and intersecting it with an integral grid, assigning a when the ray intersects a horizontal grid line and b when it meets a verticalgrid line. Indeed, the slope of a Sturmian sequence is the density of a in thesequence (one shows that the limit as n tends to infinity of the average ofthe number of occurrences a in u · · · u n exists, see [4], Proposition 6.1.10). Lemma 2.2.
Let u be a Sturmian word whose slope has an unboundedregular continued fraction expansion. Then, for every positive integer n ,there are finite words U , V and a positive real number s such that U V s isa prefix of u and | U V s | ≥ n | U V | .Proof. This follows from the proof of Proposition 11.1 from [3]. (cid:3)
Remark 4.
We apply the above lemma to Sturmian sequences where both a, b are of the form ( ε, r ), with ε = ± r ∈ N . In particular, we use thisin the context of Rosen expansions to prove Corollary 4.1.3. Bounding the height of convergents
In what follows, we fix λ = λ m for some m >
3, and suppose that ξ ∈ (0 , λ/
2) is a real algebraic number having an infinite Rosen continuedfraction expansion over Q ( λ ) . Our goal is to estimate the naive height H ( p n /q n ) of the n th convergent p n /q n . In light of Theorem 2.1, we let n be the least value of n such that q n > Lemma 3.1.
Let c = c ( λ ) be defined by c = min σ | σ ( λ ) | λ , where theminimum is taken over all field embeddings of Q ( λ ) into R . Then for all n ≥ n , and any such σ , we have both q n ≥ c | σ ( q n ) | and p n ≥ c | σ ( p n ) | . Proof.
For any n ≥ n , recall that M n = (cid:18) p n − p n q n − q n (cid:19) ; this is clearly anelement of ∆ m . By Theorem 2.1 we have q n + p n − ≥ | σ ( q n + p n − ) | .Now let j ∈ N and set M n,j = (cid:18) p n − p n q n − q n (cid:19) (cid:18) jλ (cid:19) = (cid:18) p n − p n + jλp n − q n − q n + jλq n − (cid:19) . This is also an element of ∆ m of trace greater than 2, and hence | p n − + q n + jλq n − | ≥ | σ ( p n − + q n ) + jσ ( λq n − ) | . Since this holds for all positive j , we must have that λq n − ≥ | σ ( λq n − ) | .That is, q n − ≥ | σ ( λ ) | λ | σ ( q n − ) | ≥ (cid:18) min σ | σ ( λ ) | λ (cid:19) | σ ( q n − ) | . Similarly, using N n,j = (cid:18) p n − p n q n − q n (cid:19) (cid:18) jλ (cid:19) = (cid:18) p n − + jλp n p n q n − + jλq n q n (cid:19) , we find p n ≥ | σ ( λ ) | λ | σ ( p n ) | ≥ (cid:18) min σ | σ ( λ ) | λ (cid:19) | σ ( p n ) | . (cid:3) RANSCENDENCE WITH ROSEN CONTINUED FRACTIONS 11
Remark 5.
We conjecture that in fact q n is always greater than or equalto its conjugates, thus that in the above one can replace c by 1. Lemma 3.2.
Let D denote the field extension degree [ Q ( λ ) : Q ] . Thereexists a constant c = c ( λ ) such that for all n ≥ n , H ( p n /q n ) ≤ c q Dn . Proof.
Since p n and q n are algebraic integers of degree at most D , it followsfrom Lemma 3.1 that h ( p n /q n ) ≤ X σ D log max {| σ ( p n ) | , | σ ( q n ) |} ≤ c ′ + log q n , where σ runs through the complex embeddings, for a suitable positive con-stant c ′ . Using (7), we get the asserted estimate. (cid:3) Lemma 3.3.
Let α be a real number in [ − λ/ , λ/ with an ultimatelyperiodic expansion in Rosen continued fraction. Denote by ( p n /q n ) n ≥ thesequence of its convergents. Denote by µ the length of the preperiod and by ν the length of the period, with the convention that µ = 0 if the expansion ispurely periodic. Then α is of degree at most over Q ( λ ) , and there exists c = c ( λ, α ) such that H ( α ) ≤ c ( q µ q µ + ν ) D . Proof.
In the notation of Equation (5), α is fixed by M = M − µ M µ + ν . It thussatisfies a quadratic equation with entries in Z [ λ ], and hence is of degreeat most 2 over Q ( λ ). Indeed, α is a root of f ( x ) = cx + ( d − a ) x − b with a, b, c, d denoting the entries of M . Each entry is a Z -linear combination ofmonomials of the form rs with r an entry of M µ and s an entry of M µ + ν .Now, α is also a root of ˜ f ( x ) = Q σ σ ( f )( x ) ∈ Z [ x ], where σ ( f ) denotesthe result of applying σ to the coefficients of f ( x ). By Lemma 3.1, all of theconjugates of each of p µ , p µ − , q µ − , q µ can be bounded by the product of q µ with a constant depending upon α and λ . Similarly for the entries of M µ + ν .After some computation, we conclude that the height of α is ≪ q Dµ q Dµ + ν .(One checks that the case of µ = 0 is subsumed by the above.) (cid:3) Remark 6.
Whereas a real number whose regular continued fraction expan-sion is ultimately periodic is exactly of degree two over the field of rational numbers, in the previous lemma the words “at most” are necessary. Indeed, x = 1 has an ultimately periodic Rosen expansion with respect to any λ m with m even, [20]. Further examples of elements of Q ( λ m ) with periodicexpansions are easily given when m ∈ { , } , see Corollary 1 of [23]. Yetfurther examples, including cases with m ∈ { , } , are given in [21], [13].4. Transcendence results
As usual, ≪ and ≫ denote inequality with implied constant.4.1. Applying Roth–LeVeque: the proof of Theorem 1.1.
We nowshow that the sequence of denominators of convergents to an algebraic num-ber cannot grow too quickly. Theorem 1.1 then follows.
Proof.
Let ε be a positive real number. Let ζ be an algebraic number havingan infinite Rosen expansion with convergents r n /s n .By the Roth–LeVeque Theorem 2.2, we have | ζ − r n /s n | ≫ H ( r n /s n ) − − ε , for n ≥ . And, hence by Lemma 3.2, for n ≥ n = n ( ζ ), we have | ζ − r n /s n | ≫ s − D − Dεn . Inequality (6) then gives that there exists a constant c (indepen-dent of n ≥ n ) such that s n +1 < c s D − Dεn . Set a = 2 D − Dε . For j < n , define ℓ j such that s j < ℓ j s aj − . Weset c = max { , c , ℓ , . . . , ℓ n − } and find that for any n > s n +1 < c s an < c ( c s an − ) a ≤ ( c s n − ) a and continuing in this manner, we have s n +1 < ( c s ) a n . Since s n +1 > s n letting c = c s , gives log s n < a n log c . From this follows thatlim sup n → + ∞ log log s n n < log( D (2 + ε ) − . Letting ε go to zero, we see that every algebraic number satisfieslim sup n → + ∞ log log s n n ≤ log(2 D − , as asserted. (cid:3) RANSCENDENCE WITH ROSEN CONTINUED FRACTIONS 13
Proof of Theorem 1.2 and an application.
Proof.
With λ = 2 cos π/m fixed, given ξ of infinite Rosen continued fractionwith convergents ( p n /q n ) n ≥ , we let b = lim inf n q /nn and B = lim sup n q /nn ,and assume that B < ∞ . Let η be a positive real number with b − < η < b .Since there are only finitely many n with either q /nn < b − η or q /nn > B + η ,we have both that q n ≫ ( b − η ) n and q n ≪ ( B + η ) n .Suppose that w is a positive real number and U , V are finite words in {± } × N such that U V w is a prefix of the infinite word composed of thepartial quotients of ξ . Denote by α the real number of degree at most twoover Q ( λ ) whose Rosen continued fraction is given by the word U V ∞ , where V ∞ means the concatenation of infinitely many copies of V . Set | U | = u and | V | = v . Since ξ and α have their first ⌊ u + vw ⌋ partial quotients incommon, we have | ξ − α | < c q − ⌊ u + vw ⌋ ≪ ( b − η ) − u + vw ) . Furthermore, it follows from Lemma 3.3 that H ( α ) ≪ ( q u q u + v ) D ≪ ( B + η ) D (2 u + v ) . Combined with the previous inequality, this gives | ξ − α | ≪ H ( α ) − u + vw ) log( b − η ) / ( D (2 u + v ) log( B + η )) . Now suppose that ξ is algebraic of degree greater than two over Q ( λ ).Then, for every ε >
0, there exists a positive constant C ( ε ) such that everyreal algebraic number β of degree at most 2 over Q ( λ ) satisfies | ξ − β | > C ( ε ) H ( β ) − − ε . This follows from Theorem 2.2 if β is in Q ( λ ) and, otherwise, by applyingTheorem 2.4 with t = 2 and dt = δ to each subfield K of Q ( λ ).This proves that ξ must be transcendental if there are u, v, w such that u + vw is arbitrarily large and2( u + vw ) log bD (2 u + v ) log B > , as asserted. (cid:3) Corollary 4.1.
A Rosen continued fraction whose sequence of partial quo-tients is Sturmian with slope of unbounded regular continued fraction partialquotients represents a transcendental number.Proof.
Combine Lemma 2.2 with the previous Theorem. (cid:3)
Remark 7.
Using the Subspace Theorem as in [1] does not yield in generalan improvement of Theorem 1.2. In case u = 0 , b = B , inequality (1)reduces to w > D/
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