Rational approximation to values of G-functions, and their expansions in integer bases
aa r X i v : . [ m a t h . N T ] O c t Rational approximation to values of G -functions, andtheir expansions in integer bases S. Fischler and T. RivoalAugust 22, 2018
Abstract
We prove a general effective result concerning approximation of irrational values atrational points a/b of any G -function F with rational Taylor coefficients by fractionsof the form n/ ( B · b m ), where the integer B is fixed. As a corollary, we show thatif F is not in Q ( z ), for any ε > b and m large enough with respect to a , ε and F , then | F ( a/b ) − n/b m | ≥ /b m (1+ ε ) and F ( a/b ) / ∈ Q . This enables us toobtain a new and effective result on repetition of patterns in the b -ary expansion of F ( a/b s ) for any b ≥
2. In particular, defining N ( n ) as the number of consecutiveequal digits in the b -ary expansion of F ( a/b s ) starting from the n -th digit, we provethat lim sup n N ( n ) /n ≤ ε provided the integer s ≥ b s is large enoughwith respect to a , ε > F . This improves over the previous bound 1 + ε , thatcan be deduced from the work of Zudilin.Our crucial ingredient is the use of non-diagonal simultaneous Pad´e type approx-imants for any given family of G -functions solution of a differential system, in aconstruction `a la Chudnovsky-Andr´e. This idea was introduced by Beukers in theparticular case of the function (1 − z ) α in his study of the generalized Ramanujan-Nagell equation, and we use it in its full generality here. In contrast with the classicalDiophantine “competition” between E -functions and G -functions, similar results arestill not known for a single transcendental value of an E -function at a rational point,not even for the exponential function. This paper deals with approximations of values of G -functions at rational points by rationalnumbers with denominator a power of a fixed integer; an important motivation is thatperiods are conjecturally values of G -functions (see [18, Section 2.2]). Before stating ourresults, we recall some important results in the Diophantine theory of G -functions, as wellas of E -functions, even though no new result will be given for the latter. Throughout thepaper we fix an embedding of Q into C . 1 efinition 1. A G -function F is a power series F ( z ) = P ∞ n =0 a n z n such that the coeffi-cients a n are algebraic numbers and there exists C > such that, for any n ≥ : ( i ) the maximum of the moduli of the conjugates of a n is ≤ C n . ( ii ) there exists a sequence of rational integers d n , with | d n | ≤ C n , such that d n a m isan algebraic integer for all m ≤ n . ( iii ) F ( z ) satisfies a homogeneous linear differential equation with coefficients in Q ( z ) .An E -function is a power series F ( z ) = P ∞ n =0 a n n ! z n such that P ∞ n =0 a n z n is a G -function. Siegel’s original definition [24] of E and G -functions is slightly more general but it isbelieved to define the same functions as above. It is a fact that the Diophantine theory of G -functions is not as fully developped as that of E -functions. There is no general theoremabout the transcendence of values of G -functions, but results like the following one, due toChudnovsky [16]. Let N ≥ and Y ( z ) = t ( F ( z ) , . . . , F N ( z )) be a vector of G -functions solution of adifferential system Y ′ ( z ) = A ( z ) Y ( z ) , where A ( z ) ∈ M N ( Q ( z )) . Assume that F ( z ) , . . . , F N ( z ) are C ( z ) -algebraically independent. Then for any d , there exists C = C ( Y, d ) > such that, for any algebraic number α = 0 of degree d with | α | < exp( − C log ( H ( α )) N N +1 ) , there does not exist a polynomial relation between the values F ( α ) , . . . , F N ( α ) over Q ( α ) of degree d . Here, H ( α ) is the naive height of α , i.e. the maximum of the modulus of the coefficientsof the (normalized) minimal polynomial of α over Q . Chudnovsky’s theorem refines theworks of Bombieri [12] and Galochkin [19]. Andr´e generalized Chudnovsky’s theorem to thecase of an inhomogenous system Y ′ ( z ) = A ( z ) Y ( z ) + B ( z ), A ( z ) , B ( z ) ∈ M N ( Q ( z )), witha similar condition on α and H ( α ); see [4, pp. 130–138] when the place v is archimedean.Andr´e and Chudnovsky’s theorems are still essentially the best known today in this gen-erality but they are far from being transcendence or algebraic independence statements.We recall that, in fact, it is not even known if there exist three algebraically independent G -values. ( ) On the other hand, the situation is best possible for E -functions, by theSiegel-Shidlovsky Theorem [24, 23]: except maybe for a finite set included in the set ofsingularities of a given differential system satisfied by E -functions, the numerical transcen-dence degree over Q of the values of the latter at a non-zero algebraic point is equal to theirfunctional transcendence degree over Q ( z ). Beukers [11] was even able to describe veryprecisely the nature of the numerical algebraic relations when the transcendence degree isnot maximal.A lot of work has been devoted to improvements of Chudnovsky’s theorem, or alike, forclassical G -functions like the polylogarithms P ∞ n =1 z n /n s , or to determine weaker conditionsfor the irrationality of the values of G -functions at rational points. From a qualitative pointof view, the result is the following. There exist examples of two algebraically independent G -values, for instance π and Γ(1 / , or π andΓ(1 / . This was first proved by Chudnovsky with a method not related to G -functions, but Andr´e [5]obtained a proof with certain Gauss’ hypergeometric functions, which are G -functions. Andr´e’s method isvery specific and has not been generalized. et F be a G -function with rational Taylor coefficients such that F ( z ) Q ( z ) . Thenthere exist positive constants C and C , depending only on F , with the following property.Let a = 0 and b ≥ be integers such that b > ( C | a | ) C . (1.1) Then F ( a/b ) is irrational. This result follows from Theorem I in [16, 17], together with an irrationality measure;see also [19]. This measure and the value of C have been improved by Zudilin [26], underfurther assumptions on F . He obtains the following result (in a more precise form). Theorem 1 (Zudilin [26]) . Let N ≥ and Y ( z ) = t ( F ( z ) , . . . , F N ( z )) be a vector of G -functions solution of a differential system Y ′ ( z ) = A ( z ) Y ( z ) + B ( z ) , where A ( z ) , B ( z ) ∈ M N ( C ( z )) . Assume either that N = 2 and , F ( z ) , F ( z ) are C ( z ) -linearly independent,or that N ≥ and F ( z ) , . . . , F N ( z ) are C ( z ) -algebraically independent. Let ε > , a ∈ Z , a = 0 . Let b and q be sufficiently large positive integers, in terms of the F j ’s, a and ε ; then F j ( a/b ) is an irrational number and for any integer p , we have (cid:12)(cid:12)(cid:12)(cid:12) F j (cid:16) ab (cid:17) − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ q ε , j = 1 , . . . , N. (1.2)Zudilin’s proof follows Shidlovsky’s ineffective approach to zero estimates (see [23, p. 93,Lemma 8]). It is likely that using an effective method instead (see [4, Appendix of Chap-ter III], [9] and [15]), one would make Theorem 1 effective. We mention that Zudilin [25]also obtained similar irrationality measures for the values of E -functions at any non-zerorational point.We now come to our main result. Roughly speaking, it is an improvement of Zudilin’sexponent 2 + ε in (1.2) when q is restricted to integers of the form b m . In this case, theexponent drops from 2 + ε to 1 + ε ; see Corollary 1 with B = 1. We first state a moreprecise and general version, without ε , which contains an irrationality measure in disguise(see the comments following the theorem). Theorem 2.
Let F be a G -function with rational Taylor coefficients and with F ( z ) Q ( z ) ,and t ≥ . Then there exist some positive effectively computable constants c , c , c , c ,depending only on F (and t as well for c ), such that the following property holds. Let a = 0 and b, B ≥ be integers such that b > ( c | a | ) c and B ≤ b t . (1.3) Then F ( a/b ) Q , and for any n ∈ Z and any m ≥ c b )log( | a | +1) we have (cid:12)(cid:12)(cid:12) F (cid:16) ab (cid:17) − nB · b m (cid:12)(cid:12)(cid:12) ≥ B · b m · ( | a | + 1) c m . (1.4)3n the case of the dilogarithm Li ( z ) = P ∞ n =1 z n n , our proof provides c = 4 e , c = 12and c = 10 . We did not try to compute c because it is useless for the application statedin Theorem 3 below, but this could be done in principle. Needless to say, these valuesare far from best possible but this is not the point of this paper. For related results, butrestricted only to the G -functions (1 − z ) α and log(1 − z ), see [7, 8, 10] and [21] respectively.We point out that Theorem 2 is effective, because an effective zero estimate (due to Andr´e)is used. In contrast with Zudilin’s theorem, we only need to assume that F ( z ) Q ( z ).The lower bound (1.4) implies an effective irrationality measure of F ( a/b ). Indeed, let A and B ≥ t = log( B )log( b ) and m = ⌊ c b )log( | a | +1) ⌋ + 1. The proof of Theorem 2shows that one may take c = t if t (i.e. B with our choice here) is large enough in termsof F . Then, with n = A · b m , Eq. (1.4) implies that, provided b > ( c | a | ) c , we have (cid:12)(cid:12)(cid:12)(cid:12) F (cid:16) ab (cid:17) − AB (cid:12)(cid:12)(cid:12)(cid:12) ≥ κB µ (1.5)for some constants κ, µ > a , b , and F . The constant µ isworse that Zudilin’s, at least when b is large with respect to a , but (1.5) applies to alarger class of G -functions. On the other hand, when F ( z ) Q ( z ), we can compare (1.5)with Chudnovsky’s irrationality measure [16, 17] under assumption (1.1): our constant c = 3( N + 2) in (1.3) is slightly worse than his C = N ( N + 1) /ε + N + 1 (when his ε > C , which depends on F and ε . In any case, it is difficult to compare such results in the literature as they apply tomore or less G -functions, to more or less values a/b , and give more or less close to optimalirrationality measures.We now extract a lower bound similar to Zudilin’s measure (1.2). Given ε > m ≥ t/ε and b > ( | a | + 1) c /ε , we derive the following corollary fromTheorem 2. It is also effective and it will be used to prove Theorem 3 below. Corollary 1.
Let F be a G -function with rational Taylor coefficients and with F ( z ) Q ( z ) , ε > , t ≥ and a ∈ Z , a = 0 . Let b and m be positive integers, sufficiently large in termsof F , ε , a (and t for m ). Then F ( a/b ) Q and for any integers n and B with ≤ B ≤ b t ,we have (cid:12)(cid:12)(cid:12) F (cid:16) ab (cid:17) − nB · b m (cid:12)(cid:12)(cid:12) ≥ b m (1+ ε ) . We don’t know if some analogues of Theorem 2 and Corollary 1 hold when F is supposedto be an E -function. This is surprising because, as we indicated above, the Diophantinetheory of E -values is much more advanced than that of G -values. Our method is inoperantfor E -functions and we could not find any way to fix it. We explain the reason for thisunusual advantage of G -functions in the final Section 5. We also explain there that ananalogue of Theorem 2 holds for 1 /F ( z ) instead of F ( z ) under a less general assumptionon the G -function F ( z ).The quality of restricted rational approximants as in Theorem 2 and Corollary 1 can bemeasured (when t = 0) by a Diophantine exponent v b studied in [3]. Given ξ ∈ R \ Q , v b ( ξ )4s the infimum of the set of real numbers µ such that | ξ − nb m | ≥ b − m (1+ µ ) for any n ∈ Z andany sufficiently large m . With this notation, the special case t = 0 of Corollary 1 reads v b ( F ( a/b )) ≤ ε . The metric properties of this Diophantine exponent are studied in [3,Section 7]: with respect to Lebesgue measure, almost all real numbers ξ satisfy v b ( ξ ) = 0for any b ≥
2, and given b ≥ ξ such that v b ( ξ ) ≥ ε has Hausdorff dimension ε . Therefore Theorem 2 and Corollary 1 are a step towards the conjecture that valuesof G -functions behave like generic real numbers with respect to rational approximation.Our results have interesting consequences on the nature of the b -ary expansions ofvalues of G -functions; this is a class of numbers for which very few such results are known(see [13, 14]). Let b , t be integers with b ≥ t ≥
1, and let ξ ∈ R \ Q . We denote by0 .a a a . . . the expansion in base b of the fractional part of ξ . For any n ≥
1, let N b ( ξ, t, n )denote the largest integer ℓ such that ( a n a n +1 . . . a n + t − ) ℓ is a prefix of the infinite word a n a n +1 a n +2 . . . . In other words, it is the number of times the pattern a n a n +1 . . . a n + t − isrepeated starting from a n . Obviously N b ( ξ, t, n ) ≥
1, and N b ( ξ, t, n ) is finite since ξ isirrational. If t = 1, N b ( ξ, t, n ) is simply the number of consecutive equal digits in theexpansion of ξ , starting from a n . For almost all real numbers ξ with respect to Lebesguemeasure, lim n →∞ n N b ( ξ, t, n ) = 0. Theorem 3.
Let F be a G -function with rational Taylor coefficients and with F ( z ) Q ( z ) , ε > , and a ∈ Z , a = 0 . Let b ≥ . Then for any s ≥ such that b s is sufficiently largein terms of F , ε , and a , we have for any t ≥ : lim sup n →∞ n N b (cid:0) F ( a/b s ) , t, n (cid:1) ≤ ε/t. In the case of the dilogarithm, this result applies to Li (1 /b s ) for a = 1, any fixed ε ∈ (0 , t ≥ b ≥ s ≥ /ε . A similar bound on thisupper limit, but with 1 + ε instead of ε , follows from (and under the assumptions of)Theorem 1. Conjecturally, we have lim n n N b ( ξ, t, n ) = 0 whenever ξ is a transcenden-tal value of a G -function, but it seems that the only such ξ for which the upper boundlim sup n n N b ( ξ, , n ) < ξ is an irrational algebraic number, Ridout’s theorem [20] yields v b ( ξ ) = 0 andlim n n N b ( ξ, t, n ) = 0 for any b and any t . It is not effective: for a general real alge-braic number ξ , given b , t and ε >
0, no explicit value of N ( ξ, b, t, ε ) is known such that N b ( ξ, t, n ) ≤ εn for any n ≥ N ( ξ, b, t, ε ). On the contrary, if ξ = F ( a/b s ) then Theorem 3(which is effective) provides such an explicit value provided b s is large enough – recall thatalgebraic functions which are holomorphic at 0 are G -functions. However if ξ is fixed thenTheorem 3 applies only if ε is not too small: we do not really get an effective versionof Ridout’s theorem for ξ . For other results concerning the b -ary expansions of algebraicnumbers, we refer the reader to [2, 6, 22].We proved in [18] that any real algebraic number is equal to F (1) for some algebraic G -function F with rational coefficients and radius of convergence arbitrarily large. Unfor-tunately, we do not have a control on the growth of the sequence of denominators of the5oefficients of F , which is important in the computation of the constants in Theorem 2.Therefore, we cannot prove that any real algebraic number can be realized as a G -value F ( a/b ) to which Theorem 2 applies.Finally, let us explain the basic reason behind our improvement on Zudilin’s exponent.To estimate the difference | F ( ab ) − pq | using the methods of this article, we need at somepoint to find a lower bound on a certain difference D = | pq − ub k v | between two distinctrationals ( k ∈ N = { , , , . . . } , p, q, u, v ∈ Z ). When q could be anything, the best we cansay is that, trivially, D ≥ ( b k qv ) − ; however, if we know in advance that q = B · b m thenwe can improve the trivial bound to D ≥ ( b max( m,k ) Bv ) − and we save a factor b min( m,k ) inthe process. The fraction ub k v is obtained by constructing (inexplicit) Pad´e approximantsof type II to F ( z ) and the other G -functions appearing in a differential system of order 1satisfied by F . Inexplicit Pad´e approximation is a classical tool in the Diophantine theoryof G -functions.Our main new ingredient is the use of non-diagonal Pad´e type approximants, i.e. thepolynomials are made to have different degrees, which creates the factor b k we need. Thisidea seems to have been introduced in [10] in a particular case; we use it in its full generality.To illustrate its importance for Theorem 3, we remark that if one tries to compute anirrationality measure for F ( a/b ) under the assumptions of Eq. (1.1) with the method ofthe present paper, one gets an irrationality exponent not smaller than N + 1 + ε , where N is the least integer such that 1 , F ( z ) , . . . , F ( N ) ( z ) are linearly dependent over Q ( z ).The structure of this paper is as follows. Section 2 is devoted to general results onPad´e type approximation, and Section 3 to the proof of Theorem 2. At last, we deduceTheorem 3 in Section 4 and conclude with some remarks in Section 5. We gather in this section known results and preparatory computations that will be usedin Section 3 to prove Theorem 2.
Let F ( z ) , . . . , F N ( z ) be G -functions with rational coefficients. We let F ( z ) = 1 and assumethat F , F , . . . , F N are linearly independent over Q ( z ). We assume also that Y ( z ) = t (cid:0) , F ( z ) , . . . , F N ( z ) (cid:1) is a solution of a differential system of order 1 Y ′ ( z ) = A ( z ) Y ( z ) (2.1)where A ( z ) ∈ M N +1 ,N +1 ( Q ( z )) is a matrix of which the rows and columns are numberedfrom 0 to N .Let D ( z ) be a non-zero polynomial in Z [ z ] such that D ( z ) A ( z ) ∈ M N +1 ,N +1 ( Z [ z ]). Let d ∈ N be such that deg D ( z ) ≤ d and deg D ( z ) A i,j ( z ) ≤ d − A i,j ( z ) of A ( z ). We observe that D ( z ) is not a constant polynomialbecause if A ( z ) has polynomial entries, the system (2.1) cannot have a non-zero vector ofsolutions consisting of G -functions; therefore d ≥ p, q, h such that p ≥ q ≥ N h ≥
0, we can find N polynomials P ( z ) , . . . , P N ( z ) ∈ Q [ z ] of degree ≤ p and Q ( z ) ∈ Q [ z ] of degree ≤ q , such that theorder at z = 0 of R j ( z ) := Q ( z ) F j ( z ) − P j ( z )is at least p + h + 1 for all j = 1 , . . . , N . In particular, Q ( z ) is not identically zero. Wesay that ( Q ; P , . . . , P N ) is a Pad´e type approximant of type II [ q ; p, . . . , p ; p + h + 1] of( F , . . . , F N ). It is not unique in general.In what follows it is convenient to let P ( z ) = Q ( z ) and R ( z ) = 0, even though theydo not play exactly the same role as the other P j ’s and R j ’s.Set P ( z ) = t (cid:0) P ( z ) , . . . , P N ( z ) (cid:1) and R ( z ) = t (cid:0) R ( z ) , . . . , R N ( z ) (cid:1) . Following Chud-novsky [16, 17], for k ≥ P k ( z ) := t (cid:0) P ,k ( z ) , . . . , P N,k ( z ) (cid:1) ∈ Q [ z ] N +1 and R k ( z ) := t (cid:0) R ,k ( z ) , . . . , R N,k ( z ) (cid:1) ∈ Q [[ z ]] N +1 by P k ( z ) := 1 k ! D ( z ) k (cid:16) ddz − A ( z ) (cid:17) k P ( z ) , (2.2) R k ( z ) := 1 k ! D ( z ) k (cid:16) ddz − A ( z ) (cid:17) k R ( z ) . Now recall that F = 1 , F , . . . , F N are linearly independent, so that the matrix A ( z )is uniquely determined by these functions and the zero-th row of A is identically zero.Therefore we obtain the formula Q k ( z ) = 1 k ! D ( z ) k Q ( k ) ( z ) (2.3)where Q k ( z ) := P ,k ( z ). An important property is that if Q ( z ) ∈ Z [ z ], then Q k ( z ) ∈ Z [ z ]for any k because k ! ( z j ) ( k ) = (cid:0) jk (cid:1) z j − k . Moreover we have for any k ∈ N and any j :deg Q k ≤ q + ( d − k and deg P j,k ≤ p + ( d − k. We shall make use of the following results. Part ( i ) follows easily from the bounds onthe degrees of Q k and P j,k and the relation R j,k = Q k F j − P j,k : see [17, § ii ) isthe difficult one: it is a refinement and correction by Andr´e [4, p. 115] of Chudnovsky’szero estimate [16, 17]. The fact that F , . . . , F N are G -functions is used only to make theconstant in ( ii ) effective. Theorem 4 (Chudnovsky, Andr´e) . Let ( Q ; P , . . . , P N ) be a Pad´e type approximant oftype II [ q ; p, . . . , p ; p + h + 1] of ( F , . . . , F N ) ; recall that F ( z ) = 1 , F ( z ) , . . . , F N ( z ) are Q ( z ) -linearly independent G -functions with rational coefficients. Then: ( i ) For any k ≥ such that h ≥ kd , ( Q k ; P ,k , . . . , P N,k ) is a Pad´e type approximant [ q + k ( d − p + k ( d − , . . . , p + k ( d − p + h + 1 − k ]7 f ( F , . . . , F N ) . ( ii ) The determinant ∆ N ( z ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( z ) · · · Q N ( z ) P , ( z ) · · · P ,N ( z ) ... ... ... P N, ( z ) · · · P N,N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is not identically zero provided h ≥ h , where h is a positive constant, which depends onlyon F , . . . , F N and can be computed explicitly. Let us deduce precisely this result from Andr´e’s theorem. Given distinct integers i, j ∈{ , . . . , N } we haveord ( P i F j − P j F i ) = − ord ( Q ) + ord ( P i ( QF j − P j ) − P j ( QF i − P i )) ≥ − ord ( Q ) + min(ord ( P i ) , ord ( P j )) + p + h + 1 ≥ p + h + 1because ord ( P i ) ≥ ord ( Q ) for any i . Since we also have ord ( QF j − P j ) ≥ p + h + 1,Andr´e’s zero estimate [4, p. 115] applies as soon as h is greater than the constant he denotesby c (Λ), that we call h here. Moreover h is effective: see [4, Exercise 2, p. 126]. Forfuture reference, we notice that ( i ) and ( ii ) can be combined as soon as h ≥ max( h , N d );this will be the case below. Let us explain precisely now the construction of the P j ’s and of Q . First of all, we set F j ( z ) = ∞ X n =0 f j,n z n . Since the F j ’s are G -functions, there exist a sequence of integers d n > D > d n f j,n ∈ Z and d n ≤ D n +1 , and also a constant C > | f j,n | ≤ C n +1 for all n ≥ j . Let us write P j ( z ) = P pn =0 u j,n z n for 1 ≤ j ≤ N and Q ( z ) = P qn =0 v n z n . By definition of the P j ’s and of Q , we have the equations q X k =0 f j,n − k v k = 0 , n = p + 1 , . . . , p + h, j = 1 , . . . , N (2.4)and min( n,q ) X k =0 f j,n − k v k = u j,n , n = 0 , . . . , p, j = 1 , . . . , N. (2.5)8ultiplying Eq. (2.4) by d p + h , we obtain a system of N h equations in the q +1 unknowns v , . . . , v q , with integer coefficients d p + h f j, , . . . , d p + h f j,p + h bounded in absolute value by( CD ) p + h +1 . Since q + 1 > N h , Siegel’s lemma (see for instance [23, Lemma 11, Chapter 3,p. 102]) implies the existence of a non-zero vector of solutions ( v , . . . , v q ) ∈ Z q +1 suchthat | v k | ≤ q ( CD ) p + h +1 ) Nhq +1 − Nh , k = 0 , . . . , q. (2.6)From Eq. (2.5), we see that d p P j ( z ) ∈ Z [ z ] for j = 1 , . . . , N . Let H ( A ) denote the maximumof the moduli of the coefficients of a polynomial A ( z ) with real coefficients. Since CD ≥ H ( Q ) ≤ q ( CD ) p + h +1 ) Nhq +1 − Nh . (2.7) Q k , P j,k and R j,k In this section, we collect some informations we shall use freely in the proof of Theorem 2.From the estimate (2.7), we can bound the coefficients of Q k ( z ) ∈ Z [ z ] for any k ≥ H ( Q k ) ≤ q +( d − k +1 H ( D ) k ( q ( CD ) p + h +1 ) Nhq +1 − Nh . (2.8)For any A, B ∈ C [ X ] we have H ( AB ) ≤ min(1 + deg A, B ) H ( A ) H ( B )so that H ( D k ) ≤ (1 + deg D ) k − H ( D ) k ≤ ( d + 1) k − H ( D ) k and H ( Q k ) ≤ c k H ( D ) k H ( Q )for any k ≥
0, where c k = 0 if k > q and, if k ≤ q , c k = min(1 + k deg D , Q ( k ) )( d + 1) k − q ≤ ( q − k )( d + 1) k − q ≤ q − k ( d + 1) k q ≤ q (cid:16) d + 12 (cid:17) k ≤ q +( d − k where the last inequality comes from the fact that d +12 ≤ d − for any positive integer d .Taking Eq. (2.7) into account, this concludes the proof of Eq. (2.8). The proof of this inequality in the published version of this paper contains a mistake, pointed out tous by Dimitri Le Meur and corrected in an erratum. R j,k ( z ) for 1 ≤ j ≤ N . Letting Q k ( z ) = P q +( d − kn =0 v ( k ) n z n andrecalling that R j,k = Q k F j − P j,k , Lemma 4 ( i ) yields R j,k ( z ) = ∞ X n = p + h +1 − k (cid:16) min( n,q + k ( d − X ℓ =0 f j,n − ℓ v ( k ) ℓ (cid:17) z n from which we deduce that, for | z | < /C : | R j,k ( z ) | ≤ H ( Q k )( q + k ( d −
1) + 1) max(1 , C ) q + k ( d − ∞ X n = p + h +1 − k C n | z | n ≤ H ( Q k )( q + k ( d −
1) + 1)1 − C | z | max(1 , C ) q + k ( d − ( C | z | ) p + h +1 − k . (2.9)Finally, for j = 1 , . . . , N , letting P j,k ( z ) = P p +( d − kn =0 u ( k ) j,n z n we have min( n,q +( d − k ) X m =0 f j,n − m v ( k ) m = u ( k ) j,n , n = 0 , . . . , p + ( d − k. It follows that d p +( d − k P j,k ( z ) ∈ Z [ z ] for all k ≥ ≤ j ≤ N . We split the proof into two parts: in Section 3.1 we prove a general (and technical) result,namely Eq. (3.5), from which Theorem 2 will be deduced in Section 3.2.
We keep the notation and assumptions of Section 2 concerning F , . . . , F N , A , D , d , C , D . Without loss of generality, we may assume that C ≥ t, x, y ∈ R and a, b, B, m, n, h ∈ Z such that b, m ≥ ≤ (cid:12)(cid:12)(cid:12) ab (cid:12)(cid:12)(cid:12) < min (cid:18) C , H ( D ) (cid:19) , ≤ B ≤ b t , < y < d , x > N + y. We also assume that h is sufficiently large; in precise terms, we assume that h ≥ max (cid:18) h , N d , mx − N − y (cid:19) where h is the constant in Lemma 4, and we shall also assume below (just before Eq. (3.4))that h is greater than some other positive constant that could be effectively computed interms of F , . . . , F N . 10e let β = b t/h and make one more assumption on these parameters, namely Eq. (3.4)below. At last, we fix an integer j ∈ { , . . . , N } . Then we shall deduce a lower bound on (cid:12)(cid:12) F j ( ab ) − nB · b m (cid:12)(cid:12) , namely Eq. (3.5).Changing z to − z in all functions F , . . . , F N , we may assume that a > z = 0 be a rational root of D ; let us write z = r /r with coprime integers r , r .Then r divides the leading coefficient of D , so that | r | ≤ H ( D ) and | z | ≥ | r | ≥ H ( D ) .Therefore a/b is not a root of D .To apply the constructions of Section 2 we let p = ⌊ xh ⌋ and q = ⌊ ( N + y ) h ⌋ , so that p ≥ q + m. Let us choose k now. The determinant ∆ N ( z ) of Lemma 4 ( ii ) has degree at most q + N p + ( d − N ( N + 1) /
2. We use the vanishing properties of Lemma 4 ( i ) (since h ≥ N d ) by susbtracting F i ( z ) times the zero-th row from the i -th row, for any 1 ≤ i ≤ N .We obtain that ∆ N ( z ) vanishes at 0 with multiplicity at least N ( p + h + 1) − N ( N + 1) / N ( z ) = z N ( p + h +1) − N ( N +1)2 e ∆ N ( z )where e ∆ N ( z ) has degree ≤ ℓ , with ℓ = q − N ( h + 1) + dN ( N + 1) /
2, and is not identicallyzero. Since a/b is different from 0, its multiplicity as a root of ∆ N ( z ) is at most ℓ .Following the proof of [17, Theorem 4.1], we deduce that the matrix Q ( a/b ) · · · Q N + ℓ ( a/b ) P , ( a/b ) · · · P ,N + ℓ ( a/b )... ... ... P N, ( a/b ) · · · P N,N + ℓ ( a/b ) has rank N + 1. Therefore we have nQ k ( a/b ) − Bb m P j,k ( a/b ) = 0 for some integer k , with k ≤ ℓ + N = q − N h + dN ( N + 1) / j is fixed in this proof.By construction of the polynomials P j,k ( z ) and Q k ( z ), there exist two integers U j,k , V k such that P j,k ( a/b ) = U j,k / ( d p +( d − k b p +( d − k ) and Q k ( a/b ) = V k /b q +( d − k . We deduce that ξ = d p +( d − k b p +( d − k (cid:0) nQ k ( a/b ) − Bb m P j,k ( a/b ) (cid:1) is a non-zero integer (since p ≥ q ). Moreover we have assumed that p ≥ q + m so that ξ isa multiple of b m , and thus | ξ | ≥ b m .
11n the other hand we have ξ = d p +( d − k b p +( d − k (cid:16) Q k ( a/b ) (cid:0) n − Bb m F j ( a/b ) (cid:1) − Bb m (cid:0) P j,k ( a/b ) − Q k ( a/b ) F j ( a/b ) (cid:1)(cid:17) so that | ξ | ≤ d p +( d − k b p +( d − k (cid:16)(cid:12)(cid:12) Q k ( a/b ) (cid:12)(cid:12) · (cid:12)(cid:12) n − Bb m F j ( a/b ) (cid:12)(cid:12) + Bb m (cid:12)(cid:12) R j,k ( a/b ) (cid:12)(cid:12)(cid:17) . Comparing this upper bound and the lower bound | ξ | ≥ b m we obtain (cid:12)(cid:12) Q k ( a/b ) (cid:12)(cid:12) · (cid:12)(cid:12) n − Bb m F j ( a/b ) (cid:12)(cid:12) ≥ d − p +( d − k b − p − ( d − k + m − Bb m (cid:12)(cid:12) R j,k ( a/b ) (cid:12)(cid:12) . (3.1)We shall prove below that under a suitable assumption (namely Eq. (3.4)) we have (cid:12)(cid:12) R j,k ( a/b ) (cid:12)(cid:12) < d − p +( d − k b − p − ( d − k B − (3.2)so that the right hand-side of Eq. (3.1) is positive, and Q k ( a/b ) = 0. Moreover Eq. (3.1)yields (cid:12)(cid:12)(cid:12) F j (cid:16) ab (cid:17) − nB · b m (cid:12)(cid:12)(cid:12) ≥ d − p +( d − k b − p − ( d − k B | Q k ( a/b ) | . (3.3)Now, recall that p = ⌊ xh ⌋ , q = ⌊ ( N + y ) h ⌋ , and k ≤ yh + dN ( N + 1)2 . Let us denote by O (1) any positive quantity that can be bounded (explicitly) in terms of F , . . . , F N ; such a bound may involve, among others, d , N , D , C or D . We recall that C, D ≥ Nhq +1 − Nh ≤ Ny . Then Eq. (2.8) yields H ( Q k ) ≤ (cid:0) N + y )+( d − y H ( D ) y ( CD ) ( x +1) N/y (cid:1) h · ( CD ( N + y ) h ) N/y · O (1)so that Eq. (2.9) provides, since
Ca/b < / (cid:12)(cid:12) R j,k ( a/b ) (cid:12)(cid:12) ≤ (cid:0) N + y )+( d − y H ( D ) y ( CD ) ( x +1) N/y C N + dy ( Ca/b ) x +1 − y (cid:1) h · ( CD ( N + dy ) h ) N/y · O (1) . Now let us assume, for simplicity, that y ≥ d . Then we have ( d − k ≤ dyh since hy ≥ N d , so that d p +( d − k b p +( d − k B ≤ (cid:0) β ( bD ) x + dy (cid:1) h . Therefore (3.2) holds if h is larger than some effectively computable constant (dependingonly on F , . . . , F N ) and if y ≥ d and 2 N + y )+( d − y H ( D ) y ( CD ) ( x +1) N/y C N + dy ( Ca/b ) x +1 − y ( bD ) x + dy β < . (3.4)12oreover, since a/b <
1, we have (cid:12)(cid:12) Q k ( a/b ) (cid:12)(cid:12) ≤ ( q + ( d − k + 1) H ( Q k ) ≤ (cid:16) N + y )+( d − y H ( D ) y ( CD ) ( x +1) N/y (cid:17) h · ( CD ( N + dy ) h ) N/y · O (1)so that Eq. (3.3) yields finally (since 1 /y ≤ d ): (cid:12)(cid:12)(cid:12) F j (cid:16) ab (cid:17) − nB · b m (cid:12)(cid:12)(cid:12) ≥ (cid:16) β − ( bD ) − x − dy − N + y ) − ( d − y H ( D ) − y ( CD ) − ( x +1) N/y (cid:17) h · h − Nd · O (1) − . (3.5)This is a very general lower bound and in the next section we will proceed to a suitablechoice of the parameters. In this section we prove Theorem 2 by applying the proof of Section 3.1 to suitable param-eters.Let F be a G -function with F ( z ) Q ( z ). Let F ( z ) = 1, and denote by N the leastpositive integer for which there exist a ( z ) , . . . , a N ( z ) ∈ Q ( z ) such that F ( N ) ( z ) = a N ( z ) F ( N − ( z ) + . . . + a ( z ) F ′ ( z ) + a ( z ) F ( z ) + a ( z ) . We have N ≥
1, and N = 1 may hold (it does for instance with F ( z ) = log(1 − z )).By construction and since F ( z ) Q ( z ), the G -functions F = 1, F = F , F = F ′ , . . . , F N = F ( N − are linearly independent over Q ( z ). We are in position to apply the resultsof Sections 2 and 3.1; as in Section 3.1 we may assume a > C ≥ c = 4 H ( D )( CD ) Nd +1 C and c = 3( N + 2) . We take c = max( h , h , h , N d , t ) where h is the constant implied in Lemma 4 ( ii ), h is the effectively computable constant defined just before Eq. (3.4), and h is anothereffectively computable constant to be defined below; these three constants depend only on F . Then we assume m ≥ c b )log( c a ) ; (3.6)this is a consequence of our assumption m ≥ c b )log( a +1) provided we choose c = c .We choose y = 14( d + 1) , x = 13 log( b )log( c a ) and h = (cid:22) mx − N − (cid:23) . Then (1.3) implies x ≥ N + 2, and (3.6) yields h ≥ c .13et us check that (3.4) holds. We notice that β ≤ b / since h ≥ c ≥ t . Since y = d +1) and x ≥ N + 1, the left hand side of (3.4) is less than2 N +1 H ( D )( CD ) Nd ( x +1) C x + N +2 D x +1 a x +1 b − / ≤ ( c a ) x +1 b − / ≤ / x and the lower bound log( b )log( c a ) ≥ x ≥ N + 2 ≥
3. Therefore all the assumptions made in Section 3.1 hold.We set c = D d ( N +2)( d +1) N +14 N +8 H ( D ) N +2)( d +1) ( CD ) N ( N +3)( d +1) N +2 . Using (3.5) and the various conditions on x and y , we readily obtain (cid:12)(cid:12)(cid:12) F (cid:16) ab (cid:17) − nB · b m (cid:12)(cid:12)(cid:12) ≥ β − h (cid:16) b x + dy c x (cid:17) − h · h − Nd · O (1) − ≥ B − ( bc ) − x +1 x − N − · m (3.7)provided h ≥ h , where h is effective and depends only on F . Now we have x + 1 x − N − N + 2 x · − N +1 x ≤ N + 2) log( c a )log( b )because x =
13 log( b )log( c a ) ≥ N + 2. It is trivial matter to check that for any u ≥ v ≥ e , we have log( uv ) ≤ u + 1) log( v ). Since c ≥ > e (because we always have H ( D ) ≥ C ≥ D ≥ u = a , v = c and we get x + 1 x − N − ≤ c log( a + 1)log b with c = 6( N + 2) log( c ). Hence, we deduce from (3.7) that (cid:12)(cid:12)(cid:12) F (cid:16) ab (cid:17) − nB · b m (cid:12)(cid:12)(cid:12) ≥ B · b m ( a + 1) c m c m ≥ B · b m ( a + 1) c m where c = log(2 c )log(2) c and c = c + log( c )log(2) . This completes the proof of Theorem 2.
Remark.
Let us compute the constants c , c and c in the case of the G -function Li . Thevector Y ( z ) = t (cid:0) , Li ( z ) , Li ( z ) (cid:1) is solution of the differential system Y ′ ( z ) = − z z Y ( z ) . Hence D ( z ) = z (1 − z ), H ( D ) = 1, d = 2, N = 2, C = 1 and D = e . With the constantsdefined in the proof just above, we obtain c = 4 e , c = 12 and c = 120177948 + 11850193 log(2) + 396 log(2) < . . c could be computed as well, but we did not try to do so because it is notimportant for the application to Theorem 3.It follows that Theorem 2 can be applied with F ( z ) = Li ( z ) when b ≥ e | a | > s such that b s ≥ e | a | > b s ≥ ( | a | + 1) c /ε . In particular, if a = 1, b ≥ < ε <
1, Theorem 3 applies toLi (1 /b s ) for any integer s ≥ . /ε. We have not tried to optimize our general constantswhich in this case could be decreased.
In this section we deduce Theorem 3 from Corollary 1 stated in the introduction.Let ξ = F ( a/b s ), q n = b n − ( b t − p n = ( b t − ⌊ b n − ξ ⌋ + a n b t − + a n − b t − + . . . + a n + t − . Then the b -ary expansion of p n q n = ⌊ b n − ξ ⌋ b n − + a n b t − + a n − b t − + . . . + a n + t − b n − ( b t − n + t N b ( ξ, t, n ) − b -ary expansion of ξ . Therefore we have (cid:12)(cid:12)(cid:12)(cid:12) ξ − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≤ b − b n + t N b ( ξ,t,n ) . Now Corollary 1 with b s for b , B = b t − m = ⌊ n − s ⌋ yields (cid:12)(cid:12)(cid:12)(cid:12) ξ − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≥ b ⌊ n − s ⌋ s (1+ ε ) . The comparison of both inequalities enables us to conclude the proof.
In Section 2, we assumed that the degrees of the polynomials satisfy p ≥ q and in fact p ≥ q + m , which was crucial to prove Theorem 2. The case q ≥ p also provides someinformations, but not in the exact situation of Theorem 4. Indeed, with the notation ofSection 2.1 the polynomials P j,k ( z ) with 1 ≤ j ≤ N depend on P ( z ), . . . , P N ( z ) and alsoon P ( z ) = Q ( z ) (see Eq. (2.2)). In order to be able to bound the degree of P j,k ( z ) interms of p only (independently of q ), we need to deduce from (2.2) a relation analogousto (2.3), namely an expression for the polynomials P j,k ( z ) in terms of P ( z ), . . . , P N ( z )only. This follows easily under the additional assumption that the zero-th row of A ( z ) isidentically zero, i.e. that t ( F , . . . , F N ) is a solution of a homogeneous linear differentialsystem. Following the same method as in the case p ≥ q , this enables us to prove thefollowing result. 15 heorem 5. Let F be a G -function with rational Taylor coefficients and t ≥ . Letus assume that F ( z ) is solution of a homogeneous linear differential equation of order N with coefficients in Q ( z ) and that , F ( z ) , F ′ ( z ) , . . . , F ( N − ( z ) are linearly independentover Q ( z ) . Then there exist some positive effectively computable constants e c , e c , e c , e c ,depending only on F (and t as well for e c ), such that the following property holds. Let a = 0 and b, B ≥ be integers such that b > ( e c | a | ) e c and B ≤ b t . (5.1) Then F ( a/b ) / ∈ Q and for any n ∈ Z and any m ≥ e c b )log( | a | +1) , we have (cid:12)(cid:12)(cid:12)(cid:12) F ( ab ) − nB · b m (cid:12)(cid:12)(cid:12)(cid:12) ≥ B · b m · ( | a | + 1) e c m . (5.2)Analogues of Corollary 1 and Theorem 3 for 1 /F ( a/b ) hold as well. These results canbe applied directly to the functions log(1 − z ) + √ − z and √ − z log(1 − z ) for instance,but not to log(1 − z ). Actually the proof of Theorem 5 (and of all other results in thispaper) can be generalized to number fields, at least to multiply B with a fixed non-zeroalgebraic number (and all implied constants would depend on this number), by replacingthe algebraic number ξ defined in Section 3.1 with its norm over the rationals. ApplyingTheorem 5 to √ − z log(1 − z ) with B multiplied by p − a/b and canceling out thisfactor shows that Theorem 2, Corollary 1 and Theorem 3 hold with 1 / log(1 − a/b ) insteadof F ( a/b ).A natural problem is to obtain an analogue of Theorem 2 when the F j ’s are E -functionsand not G -functions. With the same notations as in Section 2, the polynomials Q k wouldstill have integer coefficients, but the denominators of the coefficients of the polynomials P j,k would no longer be bounded by d p +( d − k but by ( p + ( d − k )! d p +( d − k . As the readermay check, this cancels the benefits of having non-diagonal Pad´e type approximants if wefollow the same method of proof as in Section 3. We don’t know if this problem can befixed to prove analogues of Theorems 2 and 3 for E -functions. Very few results are knownon b -ary expansions of values of E -functions (see [1], [13], [14]). From a conjectural pointof view, the situation is not clear either: values of E -functions do not behave like genericnumbers with respect to rational approximation, as the continued fraction expansion of e shows. Bibliography [1] B. Adamczewski,
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On a measure of irrationality for values of G -functions , Izv. Math. .1(1996), 91–118; translated from the russian version Izv. Ross. Akad. Nauk Ser. Mat. .1 (1996), 87–114.S. Fischler, Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud, CNRS, Universit´eParis-Saclay, 91405 Orsay, France.T. Rivoal, Institut Fourier, CNRS et Universit´e Grenoble Alpes, CS 40700, 38058 Grenoblecedex 9, FranceKeywords: G -functions, rational approximations, irrationality measure, Pad´e approxima-tion, integer base expansion.MSC 2000: 11J82 (Primary); 11A63, 11J25, 11J91 (Secondary).18 ddendum This addendum is not included in the published version of this paper.
When F ( ab ) is an algebraic irrational, Corollary 1 looks like Ridout’s Theorem foralgebraic irrational numbers, but this is not really the same. First, if F ( ab ) is an algebraicirrational and b is fixed, then it applies only if ε is not too small with respect to b , andthus we do not get an effective version of Ridout’s theorem “in base b ” for this number.Second, we don’t know if any algebraic irrational number can be represented as a value F ( ab ) to which these results apply.In this addendum, we deduce from Theorem 2 the following result, which partiallysolves these problems. Theorem 6.
Let d be a positive rational number such that √ d / ∈ Q . There exist someconstants η d > , κ d > and N d such that for any convergent αβ of the continued fractionexpansion of √ d with α, β ≥ N d , we have (cid:12)(cid:12)(cid:12) √ d − nα m (cid:12)(cid:12)(cid:12) ≥ η d α ) m and (cid:12)(cid:12)(cid:12)(cid:12) √ d − nβ m (cid:12)(cid:12)(cid:12)(cid:12) ≥ κ d β ) m for any integer n ∈ Z and any m large enough with respect to d, α, β . In particular, for any ε > (cid:12)(cid:12)(cid:12) √ d − nα m (cid:12)(cid:12)(cid:12) ≥ α m (1+ ε ) and (cid:12)(cid:12)(cid:12)(cid:12) √ d − nβ m (cid:12)(cid:12)(cid:12)(cid:12) ≥ β m (1+ ε ) provided α and β are large enough (in terms of d and ε ). Proof.
Let α, β be any positive integers such that | α − dβ | ≤ c ( d ) for some given constant c ( d ). Note that if α/β is a convergent to √ d , then | α − dβ | ≤ α + √ dββ ≤ √ d + 1so that c ( d ) = 2 √ d + 1 is an admissible value for all convergents.Let f ( x ) = √ − x . Then f ( α − dβ α ) = βα √ d. Let d = uv with positive integers u and v . We can apply Theorem 2 to F = f , a = vα − uβ and b = vα , provided that α > c c | α − dβ | c where c , c depend only on d . This inequality holds a fortiori if weassume that α ≥ ( c c ( d )) c / =: N d , which we now do. Then (cid:12)(cid:12)(cid:12)(cid:12) βα √ d − nB · ( vα ) m (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) α − dβ α (cid:19) − nB · ( vα ) m (cid:12)(cid:12)(cid:12)(cid:12) ≥ B · (1 + v | α − dβ | ) c m · ( vα ) m for any 1 ≤ B ≤ vα t , any n ∈ Z and any m ≥ c vα )log(1+ v | α − dβ | ) .19hus (cid:12)(cid:12)(cid:12)(cid:12) √ d − αnβB · v m α m (cid:12)(cid:12)(cid:12)(cid:12) ≥ αβ · B · (1 + vc ( d )) c m · ( vα ) m . Note that c depends on f and t . We now choose t = 2, so that c becomes absolute. With B = α and n = βv m n ′ (for any n ′ ∈ Z ), we get (cid:12)(cid:12)(cid:12)(cid:12) √ d − n ′ α m (cid:12)(cid:12)(cid:12)(cid:12) ≥ β · (1 + vc ( d )) c m · v m · α m . On the other hand, with B = α and n = βv m n ′ (for any n ′ ∈ Z ), we obtain (cid:12)(cid:12)(cid:12)(cid:12) √ d − n ′ α m +1 (cid:12)(cid:12)(cid:12)(cid:12) ≥ β · (1 + vc ( d )) c m · v m · α m +1 . Moreover, assuming m ≥ C ( d, α, β ) we have β · (1 + vc ( d )) c m v m ≤ δ m for some constant δ that depends only on d . Therefore combining the previous inequalitiesyields (cid:12)(cid:12)(cid:12)(cid:12) √ d − n ′ α m (cid:12)(cid:12)(cid:12)(cid:12) ≥ η d α ) m for any n ′ ∈ Z and any m ≥ C ( d, α, β ), where η d > d .We now prove the other inequality (cid:12)(cid:12)(cid:12)(cid:12) √ d − nβ m (cid:12)(cid:12)(cid:12)(cid:12) ≥ κ d β ) m . Any convergent of 1 / √ d (except maybe the first ones) is of the form β/α where α/β is aconvergent of √ d . Therefore we may apply the above result with 1 /d and β/α : we obtain (cid:12)(cid:12)(cid:12)(cid:12) √ d − n ′ β m (cid:12)(cid:12)(cid:12)(cid:12) ≥ η d β ) m . Since the map x /x is Lipschitz around √ d , we deduce the lower bound of Theorem 6by choosing an appropriate constant κ dd