A new lower bound for (3/2) k

Transcription

1 Journal de Théorie des Nomres de Bordeaux 1 (007, A new lower ound for (3/ k ar Wadim ZUDILIN Résumé. Nous démontrons que our tout entier k suérieur à une constante K effectivement calculale, la distance de (3/ k à l entier le lus roche est minorée ar 0,5803 k. Astract. We rove that, for all integers k exceeding some effectively comutale numer K, the distance from (3/ k to the nearest integer is greater than k. 1. Historical overview of the rolem Let and { } denote the integer and fractional arts of a numer, resectively. It is known [1] that the inequality {(3/ k } 1 (3/4 k for k 6 imlies the exlicit formula g(k k (3/ k for the least integer g g(k such that every ositive integer can e exressed as a sum of at most g ositive kth owers (Waring s rolem. K. Mahler [10] used Ridout s extension of Roth s theorem to show that the inequality (3/ k C k, where x min({x}, 1 {x} is the distance from x R to the nearest integer, has finitely many solutions in integers k for any C < 1. The articular case C 3/4 gives one the aove value of g(k for all k K, where K is a certain asolute ut ineffective constant. The first non-trivial (i.e., C > 1/ and effective (in terms of K estimate of the form ( (1 3 k > C k for k K, with C (11064, was roved y A. Baker and J. Coates [1] y alying effective estimates of linear forms in logarithms in the -adic case. F. Beukers [4] imroved on this result y showing that inequality (1 is valid with C for k K 5000 (although his roof yielded the etter choice C if one did not require an exlicit evaluation of the effective ound for K. Beukers roof relied on exlicit Padé aroximations to a tail of the inomial series (1 z m m ( m n0 n (z n and was later used y A. Duickas [7] and L. Hasieger [8] to derive inequality (1 with C and , resectively. The latter work also includes Manuscrit reçu le 8 juillet 005.

2 31 Wadim Zudilin the estimate (3/ k > k for k 5 using comutations from [6] and []. By modifying Beukers construction, namely, considering Padé aroximations to a tail of the series ( 1 (1 z m1 n0 ( m n m z n and studying the exlicit -adic order of the inomial coefficients involved, we are ale to rove Theorem 1. The following estimate is valid: ( 3 k > k k for k K, where K is a certain effective constant.. Hyergeometric ackground The inomial series on the left-hand side of ( is a secial case of the generalized hyergeometric series ( A0, A (3 1,..., A q q1f q B 1,..., B q z (A 0 k (A 1 k (A q k z k, k!(b 1 k (B q k where (A k Γ(A k Γ(A k0 { A(A 1 (A k 1 if k 1, 1 if k 0, denotes the Pochhammer symol (or shifted factorial. The series in (3 converges in the disc z < 1, and if one of the arameters A 0, A 1,..., A q is a non-ositive integer (i.e., the series terminates the definition of the hyergeometric series is valid for all z C. In what follows we will often use the q1 F q -notation. We will require two classical facts from the theory of generalized hyergeometric series: the Pfaff Saalschütz summation formula ( n, A, B (4 3F C, 1 A B C n 1 (C A n(c B n (C n (C A B n (see, e.g., [11],. 4, ( and the Euler Pochhammer integral for the Gauss F 1 -series ( A, B (5 F 1 C z Γ(C 1 t B1 (1 t CB1 (1 zt A dt, Γ(BΓ(C B rovided Re C > Re B > 0 (see, e.g., [11],. 0, ( Formula (5 is valid for z < 1 and also for any z C if A is a non-ositive integer. 0

3 A new lower ound for (3/ k Padé aroximations of the shifted inomial series Fix two ositive integers a and satisfying a. Formula ( yields (6 ( 3 3(1 ( 7 1 ( (1 8 ( k k 3 a1 3 1 k0 ( 1 ( k an integer 3 a1 ka ( a ν 3 a1 ν0 k0 3 (ak 3 ν (mod Z. ( k 3 (ak This motivates (cf. [4] constructing Padé aroximations to the function ( ( a ν a (7 F (z F (a, ; z z ν ν0 and alying them with the choice z 1/. ν0 (a 1 ν (a 1 ν z ν Remark 1. The connection of F (a, ; z with Beukers auxiliary series H(ã, ; z from [4] is as follows: a1 ( k F (a, ; z z ((1 a z 1 z k H(a 1, a; z. k0 Although Beukers considers H(ã, ; z only for ã, N, his construction remains valid for any ã C, N and z < 1. However the diagonal Padé aroximations, used in [4] for H(z and used elow for F (z, are different. Taking an aritrary integer n satisfying n, we follow the general recie of [5], [13]. Consider the olynomial (8 ( ( a n n, a n Q n (x F 1 a a 1 x ( ( a n 1 µ a n (x µ q µ x µ Z[x] µ n µ µ0 µ0

4 314 Wadim Zudilin of degree n. Then ( Q n (z 1 F (z ( a ν q nµ z µn z ν µ0 l0 l0 z ln µ0 µ l ν0 q nµ ( a l µ n1 r l z ln r l z ln P n (z 1 R n (z. ln Here the olynomial (10 n1 P n (x r l x nl Z[x], where r l l0 l µ0 has degree at most n, while the coefficients of the remainder R n (z r l z ln ln ( a l µ q nµ, are of the following form: ( a l µ r l q nµ µ0 ( ( ( a n 1 µ a n a l µ (1 nµ n µ µ µ0 (1 n (a n! ( n (a n 1 µ!(a l µ! (1 µ (a n 1!n!! µ (a l µ!(a n µ! µ0 (1 n (a n! (a n 1!n!! (a n 1!(a l! (n µ (a l µ (a n µ (a l!(a n! µ!(a n 1 µ0 µ (a l µ ( n (a n 1!(a l! n, a l, a n (1 3F (a n 1!(a l!n!! a n 1, a l. 1 If we aly (4 with the choice A al, B an and C al, we otain n (a n 1!(a l! ( n (n l n r l (1. (a n 1!(a l!n!! (a l n (a n n

5 A new lower ound for (3/ k 315 The assumed condition n guarantees that the coefficients r l do not vanish identically (otherwise ( n 0. Moreover, (n l n 0 for l ranging over the set n l n 1, therefore r l 0 for those l, while Finally, (11 (a n 1!(a l! r l (a n 1!(a l!n!!!/( n! (l n!/(l n! (a l!/(a l n! (a n 1!/(a n 1! (a l n!(l n! for l n. n!( n!(a l!(l n! R n (z r l z ln z n r νn z ν ln z n 1 n!( n! z n ( a n n ν0 ν0 (a n ν!(n ν! z ν ν!(a n ν! ( a n 1, n 1 F 1 a n 1. z Using the integral (5 for the olynomial (8 and remainder (11 we arrive at Lemma 1. The following reresentations are valid: Q n (z 1 (a n! (a n 1!n!( n! 1 0 t an1 (1 t n (1 z 1 t n dt and (a n! 1 R n (z (a n 1!n!( n! zn t n (1 t an1 (1 zt (an1 dt. 0 We will also require linear indeendence of a air of neighouring Padé aroximants, which is the suject of Lemma. We have (1 Q n1 (xp n (x Q n (xp n1 (x (1 n ( a n 1 a n ( a n n x. Proof. Clearly, the left-hand side in (1 is a olynomial; its constant term is 0 since P n (0 P n1 (0 0 y (10. On the other hand, Q n1 (z 1 P n (z 1 Q n (z 1 P n1 (z 1 Q n1 (z 1 ( Q n (z 1 F (z R n (z Q n (z 1 ( Q n1 (z 1 F (z R n1 (z Q n (z 1 R n1 (z Q n1 (z 1 R n (z,

6 316 Wadim Zudilin and from (8, (11 we conclude that the only negative ower of z originates with the last summand: Q n1 (z 1 R n (z ( a n 1 (1 n z n1( 1 O(z ( a n a n n ( ( a n 1 a n (1 n a n n 4. Arithmetic constituents z n( 1 O(z 1 O(1 z as z 0. We egin this section y noting that, for any rime > N, ( N N ord N! and ord N λ, where λ(x 1 {x} {x} 1 x x { 1 if x Z, 0 if x / Z. For rimes > a n, let ( { e min a n } { a n µ } { } µ (13 µ Z { } { } { } a n a µ n µ ( min a n a n µ µ µ Z a n a µ n µ ( ( min ord a n a n µ a n 0 µ n a n µ µ n µ ( ( a n 1 µ a n min ord 0 µ n µ n µ and (14 ( { } { } { } a n µ a n µ e min µ Z { } { } { } a n a µ n µ

7 min µ Z Set (15 Φ Φ(a,, n A new lower ound for (3/ k 317 ( a n µ a n µ a n a µ n µ ( ( a n µ a n. µ n µ min 0 µ n ord > an From (8, (10 and (13, (15 we deduce e and Φ Φ (a,, n > an Lemma 3. The following inclusions are valid: ( ( a n 1 µ a n Φ 1 Z µ n µ for µ 0, 1,..., n, hence Φ 1 Q n (x Z[x] and Φ 1 P n (x Z[x]. Sulementary arithmetic information for the case n relaced y n 1 is given in Lemma 4. The following inclusions are valid: (16 ( ( (n1φ 1 a n µ a n 1 Z for µ 0, 1,..., n1, µ n 1 µ hence (n 1Φ 1 Qn1 (x Z[x] and (n 1Φ 1 Pn1 (x Z[x]. Proof. Write ( ( ( a n µ a n 1 a n µ µ n 1 µ µ Therefore, if n 1 µ then (17 ( ( a n µ a n 1 ord µ n 1 µ ( a n n µ ord ( a n µ µ e. a n 1 n 1 µ. ( a n e n µ ;

8 318 Wadim Zudilin otherwise µ n 1 (mod, hence µ/ (n 1/ Z yielding ( ( a n µ a n 1 (18 ord µ n 1 µ { } { } { } a n µ a n µ { } { } { } a n 1 a µ n 1 µ { } { } { } a n 1 a n n 1 ( ( a n 1 a n a n 1 ord ord ord n 1 n n 1 ( ( a n µ a n a n 1 ord ord µ n µ n 1 e ord (n 1. µn Comination of (17 and (18 gives us the required inclusions ( Proof of theorem 1 The arameters a, and n will now deend on an increasing arameter m N in the following way: a αm, βm, n γm or n γm 1, where the choice of the ositive integers α, β and γ, satisfying α β and γ < β, is discussed later. Then Lemma 1 and Lalace s method give us (1 log R n (z C 0 (z lim m m (α β γ log(α β γ (α γ log(α γ γ log γ (β γ log(β γ γ log z max Re( γ log t (α γ log(1 t (α β γ log(1 zt 0 t 1 and (0 log Q n (z 1 C 1 (z lim m m (α β γ log(α β γ (α γ log(α γ γ log γ (β γ log(β γ max 0 t 1 Re( (α γ log t (β γ log(1 t γ log(1 z 1 t.

9 A new lower ound for (3/ k 31 In addition, from (13 (15 and the rime numer theorem we deduce that (1 log Φ(αm, βm, γm C lim m m C log Φ (αm, βm, γm lim m m ϕ(x dψ(x, ϕ (x dψ(x, where ψ(x is the logarithmic derivative of the gamma function and the 1-eriodic functions ϕ(x and ϕ (x are defined as follows: ϕ(x min ϕ(x, y, 0 y<1 ϕ (x min 0 y<1 ϕ (x, y, ϕ(x, y {(α γx} {(α γx y} {y} {(α β γx} {(α βx y} {γx y}, ϕ (x, y {(α γx y} {(α γx} {y} {(α β γx} {(α βx y} {γx y}. Lemma 5. The functions ϕ(x and ϕ (x differ on a set of measure 0, hence ( C C. Remark. The following roof is due to the anonymous referee. In an earlier version of this article, we verify directly assumtion ( for our secific choice of the integer arameters α, β and γ. Proof. First note that ϕ(x, 0 ϕ (x, 0 {0, 1}, which imlies ϕ(x {0, 1} and ϕ (x {0, 1}. From the definition we see that (x, y ϕ(x, y ϕ (x, y λ((α γx λ((α γx y. Assume that (αγx / Z; we lainly otain (x, y λ((αγxy 0, hence (x, y 0 unless y {(α γx}. Therefore the only ossiility to get ϕ(x ϕ (x is the following one: (3 ϕ(x, y 1 for y {(α γx} and ϕ(x, y 0 for y {(α γx}. Since ϕ (x, 0 1, we have {(α βx} {γx} 1; furthermore, {(α βx y} {γx y} {(α β γx} 1 if 0 y < 1 {(α βx}, 0 if 1 {(α βx} y < {γx}, 1 if {γx} y < 1,

10 30 Wadim Zudilin and {(α γx y} {y} {(α γx} { 0 if 0 y {(α γx}, 1 if {(α γx} < y < 1. The aove conditions (3 on ϕ(x, y imly 1 {(α βx} {(α γx} or, equivalently, (β γx Z. Finally, the sets {x R : (α γx Z} and {x R : (β γx Z} have measure 0, thus roving the required assertion. Our final aim is estimating the asolute value of ε k from elow, where ( 3 k M k ε k, M k Z, 0 < ε k < 1. Write k 3 in the form k 3(βm 1 j with non-negative integers m and j < 3β. Multily oth sides of ( y Φ 1 3 aj1, where Φ Φ(αm, βm, γm if n γm and Φ Φ (αm, βm, γm/(γm1 if n γm1, and sustitute z 1/: (4 Q n ( Φ 1 j ( 3 j ( 3 a1 F a, ; 1 P n ( Φ 1 3 aj1 R n ( 1 From (6, (7 we see that ( 3 j ( 3 a1 F a, ; 1 Φ 1 3 aj1. ( 3 3(1j (mod Z ( 3 k, hence the left-hand side equals M k ε k for some M k Z and we may write equality (4 in the form ( 1 (5 Q n ( Φ 1 j ε k M k R n Φ 1 3 aj1, where M k P n( Φ 1 3 aj1 Q n ( Φ 1 j M k Z y Lemmas 3 and 4. Lemma guarantees that, for at least one of n γm or γm1, we have M 0; we make the corresonding choice of n. Assuming furthermore that (6 C 0 ( 1 k C (β α log 3 < 0, from (1 and (1 we otain ( 1 R n Φ 1 3 aj1 < 1 for all m N 1,

11 A new lower ound for (3/ k 31 where N 1 > 0 is an effective asolute constant. 1 we have M k Therefore, y (5 and ( 1 Q n ( Φ 1 j ε k M k R n Φ 1 3 aj1 > 1, hence from (1, (0 we conclude that ε k > Φ j1 Q n ( Φ 3β Q n ( > em(c 1(1/C δ for any δ > 0 and m > N (δ, rovided that C 1 (1/ C δ > 0; here N (δ deends effectively on δ. Finally, since k > 3βm, we otain the estimate (7 ε k > e k(c 1(1/C δ/(3β valid for all k K 0 (δ, where the constant K 0 (δ may e determined in terms of max(n 1, N (δ. Taking α γ and β 1 (which is the otimal choice of the integer arameters α, β, γ, at least under the restriction β 100 we find that ( ( 1 1 C , C , and 1 if {x} [ 37, 1 ] [ 18 3 [ 8 37, [ 10 ϕ(x [ 16 37, 4 [ 18 [ 8 37, 7 [ 3 0 otherwise, hence 37, , , 1 37, 8 C C [ 4 37, 1 [ 6 37, 1 ] [ , 1 ] [ , 3 [ , 1 [ 3 14 [ 0 37, 5 [ 37, 3 [ , [ , 1, Using these comutations we verify (6, ( 1 C 0 C (β α log , and find that with δ e (C 1(1/C δ/(3β , 7 18 ] This result, in view of (7, comletes the roof of Theorem 1.

12 3 Wadim Zudilin 6. Related results The aove construction allows us to rove similar results for the sequences (4/3 k and (5/4 k using the reresentations ( 4 (1 ( (1 (8 3 8 ( (1 a 3a1 F a, ; 1 (mod Z, 8 and ( where 3a, ( 5 73 ( ( ( (1 a 3 a 5 3a F a, ; 3 (mod Z, 15 where 3a. Namely, taking a 5m, 15m, n 6m(1 in case (8 and a 3m, m, n 7m(1 in case ( and reeating the arguments of Section 5, we arrive at Theorem. The following estimates are valid: ( 4 k > k 3 k for k K 1, 3 ( 5 k > k 4 k for k K, 4 where K 1, K are certain effective constants. The general case of the sequence (1 1/N k for an integer N 5 may e treated as in [4] and [] y using the reresentation ( N 1 1 ( 1 1 (1 ( 1 F 0, ; (mod Z. N N 1 N 1 The est result in this direction elongs to M. Bennett []: (1 1/N k > 3 k for 4 N k3 k. Remark 3. As mentioned y the anonymous referee, our result for (4/3 k is of secial interest. It comletes Bennett s result [3] on the order of the additive asis {1, N k, (N 1 k, (N k,... } for N 3 (case N corresonds to the classical Waring s rolem; to solve this rolem one needs the ound (4/3 k > (4/ k for k 6. Thus we remain verification of the ound in the range 6 k K 1.

13 A new lower ound for (3/ k 33 We would like to conclude this note y mentioning that a stronger argument is required to otain the effective estimate (3/ k > (3/4 k and its relatives. Acknowledgements. I am very grateful to the anonymous referee for his valuale comments (incororated into the current resentation as Remarks 1 3, roviding the roof of Lemma 5 and indicating misrints. My rofound gratitude goes to Anne de Roton, for suorting my talk at the 4th Journeés Arithmetiques as a muse. I thank kindly J. Sondow for several suggestions. References [1] A. Baker, J. Coates, Fractional arts of owers of rationals. Math. Proc. Camridge Philos. Soc. 77 (175, 6 7. [] M. A. Bennett, Fractional arts of owers of rational numers. Math. Proc. Camridge Philos. Soc. 114 (13, [3] M. A. Bennett, An ideal Waring rolem with restricted summands. Acta Arith. 66 (14, [4] F. Beukers, Fractional arts of owers of rationals. Math. Proc. Camridge Philos. Soc. 0 (181, [5] G. V. Chudnovsky, Padé aroximations to the generalized hyergeometric functions. I. J. Math. Pures Al. ( 58 (17, [6] F. Delmer, J.-M. Deshouillers, The comutation of g(k in Waring s rolem. Math. Com. 54 (10, [7] A. K. Duickas, A lower ound for the quantity (3/ k. Russian Math. Surveys 45 (10, [8] L. Hasieger, Exlicit lower ounds for (3/ k. Acta Arith. 106 (003, 30. [] J. Kuina, M. Wunderlich, Extending Waring s conjecture u to Math. Com. 55 (10, [10] K. Mahler, On the fractional arts of owers of real numers. Mathematika 4 (157, [11] L. J. Slater, Generalized hyergeometric functions. Camridge University Press, 166. [1] R. C. Vaughan, The Hardy Littlewood method. Camridge Tracts in Mathematics 15, Camridge University Press, 17. [13] W. Zudilin, Ramanujan-tye formulae and irrationality measures of certain multiles of π. Mat. S. 16:7 (005, Wadim Zudilin Deartment of Mechanics and Mathematics Moscow Lomonosov State University Voroiovy Gory, GSP- 11 Moscow, Russia URL: htt://wain.mi.ras.ru/ wadim@is.ras.ru