Approximation of number π by algebraic numbers from special fields

JOURNAL

OF NUMBER

9,484

THEORY

Approximation

(1977)

of Number TT by Algebraic from Special Fields N. I.

Numbers

FELDMAN

Department of Mathematics, Faculty of Mathematics and Mechanics, University of Moscow, Moscow, V234 USSR Communicated by H. Zassenhaus Received

DEDICATED

The algebraic

estimate numbers

December

TO

from below from the

of fields

26,

PROFESSOR

the modulus generated

1973

K.

MAHLER

of the difference by the roots of unity

between is made.

v and

1

LetP(z)=a,z”+...+a,, n(m))

= n,

H(p(z))

=

oFk%%

1 uk

1,

W(z))

=

I ql

I +

***

+

I a,

I. (1)

If 5 is an algebraic number and P(z) is the irreducible polynomial with relatively prime integer coefficients, we let 40

= W(z)), H(5) = H(m),

WI

for 4

= W(z)).

(2)

Estimates from below for the modulus of the difference between rr algebraic numbers have been obtained by various authors: Popken Siegel [2], Mahler [3,4] and others. The most precise are Mahler’s equalities (see [4]) I 3-f- P/4 I > q-42, 1VT- 5 1 > H-‘I”,

and the inequalities

(3) (4)

obtained in papers [5,6]

1T - 5 1 > exp(-c&n 1n- - 5 1 > exp(-c,n

In n + 1 + In H) ln(n Inn + 2 + In H)), In(n + 2) In H),

n2 ln4(n + 2) -C In H,

48 Copyright All rights

and [l], in-

Q 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0022-314X

(5) (6)

49

APPROXIMATION OF rr

where p and q are natural numbers, c1 , c2 , c, , as well as co, c4 ,... occurring below are positive constants n = n(5), H = H(5). It is natural to compare the inequalities (3~(6) with the best possible estimate exp(-c,n In H), which it is impossible to improve (see [13]). It was shown in [7], that if m is natural, v = q(m), the Euler’s function, K = Q(exp(2+n)), 5 E K, and its degree over Q is equal to n, No = v + n-lv In(H + 2)/lnln(H + 2) E > 0, In m > E In No, then / 7T _ 5 1 > e-w&JnNo, c-5= Cs(E). (7) For n w c,v inequality (7) gives us an estimate of the form exp(-c,n In n), but as can be seen from the hypotheses of the theorem, only for 5, belonging to the fields of a special type. In the present paper a proof of some stronger result is given (in particular, the hypothesis In m > EIn No is excluded and for large v, for v M c,ZV, the v2 In v is replaced by v2). Namely, the following theorem is proved. THEOREM. Let m be integer, m 3 3, p = exp(2G/m), K = Q
N = v + vn-l In L.

5 1 > emcovN,

(8)

Note 1. It is evident that v = q(m) or 2v(m). Note 2.

y(m) > c,m(hrln m)-l (see [ll, p. 2161).

Note 3. For m bounded the inequality (8) is of the same type as inequalities (4) and (6). If n In n < v then it is worse than (6) but if n In n > v it is sharper already. It is necessary to take into consideration that there are no additional requirements to 5 in inequality (6) while inequality (8) concerns only 5 from special fields. 2

The following lemmas are needed for what follows. LEMMA 1 [5, Lemma 11. Let X be a natural nu’mber and let ai,j be real numbers, where 9i I afsj I G A

i = l,..., r;

r < t.

There exists a nontrivial collection of rational integers x1 ,..., xt for which I ais1

+ I-* + aiStx, 1 < ZAX((X I x, I < x

+ I)$/, - 2)-l,

j = l,..., t.

i = I,..., r, (9)

50

N. 1. FELDMAN

LEMMA 2 [S]. **.+a,,a,>O.

Let & ,..., 5, be all the zeros of the polynomial Then

a,zm +

a, tnl max(L i 5t I) < L(P). LEMMA 3. If p is a root of unity, the field K = Q(p, i) has the degree v, 5 E K, n(c) = n, and if the rational integers At,tena satisfy the inequality T

L

M

then either 6 = 0 or I 6 I = / t i f A,~,,i’~i~p~ t=o LO m=o

1 >, B’-uL(&‘TI”.

(10)

Proof Let K, ,..., K, be the fields conjugate to K = K1 , and &, aj be the elements of the field Kj which are conjugate to 4 = 5, and 6 = 6, . Let a, be the leading coefficient of the fundamental polynomial of 5. If 6 # 0, then

hence, applying Lemma 2, we find that 18 1 > aiT fi [ 6j 1-l > aiT ?fi (B max(1, ( [j I’))-’ j=2

3 B1-“L(ZJ-‘T’“,

for each of the roots <(I),..., C (In p/p) > In x - (l/2 In x) + E, x > 1, E = -I,3325 P
C In p -c xfl + l/(2 In x>),

x > 1.

e<=

. . ..

(11)

LEMMA 5. Suppose that s and a b 3(s + 1) are nonnegative integers. Let A, be the set of all integers obtainedfrom theproduct N(x) = (x + 1) a** (x + a) by crossing out in all possible ways no more than s of the terms in parentheses, x = 0, 3~1, &2,.... Then there is a common divisor d, of aN the numbers in A, , which satisfies the inequality a=(s + l)-Q e3*250 < d, < aa(s + l)-“.

(12)

APPROXIMATION

51

OF TT

Proof. Let p be a prime number, p < u/(s + 1). It enters in no less than e, = [u/p] parentheses of N(x). We have crossed out no more than s of them, hence the obtained number divides by peg-*. Let

d, =

n prww. P(a/(s+l)

Using Lemma 4 and the obvious inequality a!ea > CP, we obtain d, >

jJ

p-l-s+a/P

P
> exp i a In -Esfl

= exp a (

pIn P - (s + 1)

P
C

ln p

P
J

1

-

- (’ + ‘) FbT

C

2 In(u/(s + I)) + E" t1 +

2 In(u/(fL + 1)) ))

> Ua(s -i- I)-” exp(--a(1 - E + I/in 3)) 3 &(s + 1))” e-,3,25a,

ds< n P
palP = exp a (

-lnp < exp aha). P 1 s+l (

C v
LEMMA 6. Let s and Q, s < Q, be nonnegative integers, a - an arbitrary number. Then o-s C,” = c (-1)” c)u-sc:+Q-w--s. (13) tO=O

For Q = s the lemma is true. Suppose that the lemma is true ProoJ for s < Q < P. Then P--s

C,” = 1 (-l)”

c;--sc:+p--21-s

U=O P--s =

z.

(--llU

G-s(c:L-,+1

-

Cl$Lu-J

P-s+1 =

z.

(-

1Y

C%+a-a--sG+1--s

*

LEMMA 7 [lo, Lemma 61. Let n > 1, P(z) = u,zn + ..* + a,, a, # 0, u # 0. Then the polynomials P(x), P(x + a),..., P(x + nu) are linearly independent. LEMMA 8a [lo, Lemma 71. Let

R,,(z), R,(z),...,

R,(z)

be IineurZy in-

52

N. 1. FELDMAN

dependentpolynomials with degreesnot exceeding Q, let a1,..., a, be distinct nonzero numbers. Then the determinant

I RkC4a,* lk-o.I...., 9=1.2....&.

0,

x=fl,1,...,o0+a-1

is nonzero. LEMMA 8b [12, pp. 71-731. If the polynomial P(z) of degreen, + *a*+ n, - 1 satis$esthe conditions

Wak)

s = o,..., nk - 1; k = O,..., m - 1,

= b,,k ,

then m-1

n.-1

P(z) = C 1 b,,&&(z) k=O s=o

(’ ;,‘“)’

nkf-l (’ ;,“’

,

(M;-I(y))$

7=0 9l-1

M,(z) = n (z - a,p’. p=0

P#k Let a, ,..., a, be distinct nonzero numbers,the determinant E

=

1 Rk(X)

%”

k-O,l.....O. P=U.l.....P-1 2=o,i....,aO+u-1

LEMMA 9. Let E,,,,, determinant E. Then

&(X)

9

=

fi (X t=1

+

k +

1).

be the cofactor of the element R,(x) au0 of the

Eu.v.z - (E/Q!) y

(-l)“+S+zl

af-“c;4+

o-s

P-l NA>

=

(z

-

a3P

n

(z

-

G)'+~,

P=O

P5-J 9-l M,(y)

=

Y’

fl

(Y

-

a,)'+l.

P=O P#U

Proof. From Lemmas 7 and 8 we conclude that E # 0. From each of the rows of E and E,.,., we take a factor Q !. Then Eu,.u.z= (Q9a0+*-1 Fu,w 7

E = (Q!)‘“+a F,

F = I C?+tk+oa9ZI,

APPROXIMATION

OF

53

7~

and Fu,,., are the cofactors of the determinant F. Let F&z)

be the determinant obtained from F by replacing the elements of the column (u, u) by the quantities z”+Q. The derivative F:;(z) is the determinant obtained from F by replacing the elements of the column (u, V) by the quantities s!C,S+~Z”+Q-+. It is evident that F,,,(z) is a polynomial of degree qQ + Q + q - 1, which vanishes to the order Q at z = 0. Let us find its derivatives to the order Q at the points a, . By Lemma 6, c;+os! u”a’“-” = a$%!

Of” (-l)W

C;-SC~+o+(O+-W)a,“,

s < Q.

PO=0

For 0 < Q - s - w < Q, it follows that for p # u the derivative F$(a,) = 0. If s < Q - U, then w = Q - s - u is one of the numbers 0, I,..., Q - s, so subtracting from the elements of the column (u, v) of the determinant F~$(a,) quantities, which are proportional to the elements of the other columns of the group (u) we obtain the equality F~f&,) = ~!u~-“(-~)Q-~-~C~;-,F. If Q - u < s < Q then Fifi(u,) = 0, because there is no element Cf+Q+U on the right-hand side of formula (13). Thus F:;(z)

= 0,

z = 0, s = 0, l,..., Q - 1,

= 0,

z = a, , p # v, s = 0, I,..., Q, ‘+‘+Iz s! u~:-~C;+F,

= (-1)

s = 0, l,..., Q,

because for s > Q - u we have CGPS = 0. Now by Lemma 8, F,,,(z) = ‘i” (-l)‘+‘+’

s! a;-“C;-,&&,(z)

S=O

x

y

(z

-

4)r(&f;l(v))"l Y a, ;

1. I

t-0 g-1 M,(z)

=

zo

n (z P=a P7-J

-

up)Ofl.

=

aO+o+1 1 ~~+~Fu.v.. ,

It is evident that Fu.44

X=0

(’ ;,“)’

54

N.

I. FELDMAN

hence F U.B.Z = (llx!)(z-QF

I‘.V(z))‘“=’ I 0 Q-S

= F ‘f”

(-l)Q+s+u

a,?“C”,-,

a=0

C (I/r!

s!)

r=O

and the Lemma follows. 3

We proceed to the proof of the theorem. We suppose that there exist a natural constant D and 5 satisfying the hypotheses of the theorem, for which I T - [ / < exp(--2uD’iV). (14) We shall show that if the absolute constant D is sufficiently large, then this assumption leads to a contradiction. Let Q = [D3N/ln m],

q = vD2,

C = 2D3N, .x0 = DN,

so = [vD3/ln m]. (15)

We introduce the function f(z)

=

Q

o-l

c

c

&d-l

rl=o 1-O

Cdk(Z)

c,,, =

eBniZ2,

c C?&pT, r=1

0

(16)

Pk(z) = n (D3m2z + k + t),

P = ezniimP

t=1

where the rational integers Ck,l,,. satisfying 1 Ck,I,T ] < C will be chosen subsequently. Obviously, f ‘s)(x/m)

= c C,#, i C,iB(k, LZ j=O

x, ,j)(2d)8-j

pzx,

B(k, x, j) = Pf’(x/m), w,

x, i> = =

1 j,+ . .. . .jQ=j .h! j!

D3jmU

G-(4 = 1;’

j! ‘-*jQ!

c il+j’.~~;~=j ‘ ’

T = 0, T= 1.

fi (oSm2z + k + t)(‘“) 2=x/m t=l fi GJD3mx t=l

+ k + t>,

APPROXIMATION

55

OF -IT

From Lemma 5 it follows that all the integers B(k, x,j) are divided by d, , while

QQ(s + 1)-o e-3*25c < d, < QQ(s + 1)-Q,

(18)

W, x,8 = d&f@, x, s,.d,

(19)

Thus where the M(k, X, s,j) are integers and by (15) and (17) M(k,

x, s, j) < j! CQiD3h2id~1(mD3x

+ k + Q)”

-=cj! 2QD3jm2j(s + l)Q e3*25Q(2+ mD3 j x / Q-l)Q.

(20)

We set g,,,(r)

= d;lf’“‘(x/m)

= 1 C,,l 2 C,jM(k,

k.2

g,,&‘)

= 1 G,, i 7c.E

x, s, j)(2xb)“-j

p’“,

(21)

j=O

C:Wk

-y, ~,.iXXW-i

p’“.

(22)

j=O

Let us apply Lemma 1 to the system of linear forms Re gd0,

Im g&I),

x = 0, fl,...,

f (x0 - 1); S = 0, l)...) (so - I),

oft = vq(Q + 1) parameters Ck,l,r. Evidently r = 2(2x0 - 1) S, , and by virtue of (15), (20), and (22) A < q(Q + 1) IS,! 2QsoQe3s25Q(2+ Dm In rn)O i

C,‘D3’m2i(8q)so-i

j=O

d q(Q + 1) +so

exp(Q(3,25 + ln(4 + 2Dm In m)))(D3m’

Therefore there exists a nontrivial fying the inequalities I sd5)l

< I Re gs.,(5)l + I Im g,,,(5)/ < 4~C((c + l)~(O+l)~(m)/(4r-2)so

I G.z.r I < c,

+ 8q)“O-l.

(23) set of rational integers Ck,1,7 satis-

_

2)-l,

(24)

x = 0, f l,...) &(x0 - 1); s = 0, l)...) so - 1.

g,,,(5) is a polynomial with’rational integer coefficients in the algebraic numbers 5, p, i. By Lemma 2 the nonzero g5,#J satisfies an inequality of the type (10). In our case T < s and B = B,,, < q(Q + 1) ~(1 + s)O+%exp((3. 25 + In (4 + 2 I x I N-‘m In m)) Q) C(Pm2

+ 2q)‘.

(25)

56

N. 1. FELDMAN

Thus, either g,,,(5) = 0, or I sw(5)l 3 B:JZ-YS’n.

(26)

If 1x / < x0 and s < s0 , then B,,, < AC, so by virtue of (8), (15), (23)(26)

4 3 B~~L-VS/~A-1C(O+i)a~(~)/(4~~.~a)-lA-VL-V~/R~a(O+i)/(6~a~)-V >, exp(vDSN(D/8 - 1) In 2 - vDSN - v ln(2vaD5N) - 3vD3N ln(vD3)/ln m - vD3N(3,25 + ln(4 + 2Dm In m))/ln m - v2D3ln(D3m2 + 8vD2)/ln m), for v -C N and ln v -C ln(2m) then for D > D, this inequality is impossible. Hence ss.m

= 0,

x=0,&1

,..,, *(x0-

l);s=O,l,...,

so--- 1.

(27)

Let p be a rational integer. Let x, = 2pDN, FUNDAMENTAL

s, = [2*vDs In-lm].

(28)

There exists such a D, , that if for D > D,

LEMMA.

x = 0, hl,..., * (x, - 1); s = 0, l,..., s, - 1; 0 < p ,< p,, = [In D/in 21,

g,,,(5) = 0,

(29)

then

x = 0, &-l,..., f (xp+l - 1); s = 0, l)..., SP,+l - 1;

gwm = 0,

D >, D, . In addition

max

I f(y)1 < Qo exp(--4%PN),

D 3 D, .

IYlSX,+,/rn

ProoJ

(30)

By virtue of (14), (15), (18), (20), (21), (22), (28), (29)

I fYxlm>l = 4 I g,,d4l

= 4 I gs.s(4 - g,,dOl

< q(Q + 1) vC4 i

C,f I Wk

x, s, j)( (2q)*-’ I 7r - 5 1(s - j) 48-+1

1=0 <

q(Q + 1) vCd,(s + l)! (8q + Dsm38 x exp(Q(ln(2s + 2) + 3,25 + ln(2 + Dsm 1x I/Q)) - 2vD’N)

,( QQe-D’Nv, D 2 4 B DI ,

x = 0, +1,..., f (X, - 1); s = 0, l,..., (s, - 1).

(31)

APPROXIMATION

We use the Hermite interpolation

X

zfil

f

Iz--u/ml=1/(2m)

if

57

OF -H

formula

1

c\z

jsp(z

;!mi”dz.

x=1--r,

We estimate the If(y)1 for I y I < x,+,/m:

< 8(3~,)~5 xp4 max C&/(2x,)!

8xP4(3e)2rn,

(4D3m2x, + QjQ < Qo(l + 5D2m2 In m)Q < Q%T~“““(~D~)~“. Then by virtue of (15) and (31) (2s,-OS,

x

e8n”D%n

1) vC(4D3m2x,

q(Q -

If(~)1 < 2 ( ~;~D33

max lYl<~,+1/,

+ (2x, -

1) ,gQ’e-VD’N(8x,4)Sp (3e)zz~S~

< 3v2D5N 2DSN exp(3D3N + 8wD3N x

M-(

DN2*+‘-1~~vDS2”/lnm-l~

-I- 3vD’NQ’

29 QQ

exp(3D3N

exp(--vD’N

< Qo exp(-l,5vD4N

+ Q)’

49,

+

&,,D3j,,

2~)

+ vD4 ln(8D8N4) + 5vD’N) D

3

D,

2

D,

,

and the inequality (30) is proved. From this If’“’

(+)I

= 1;

$,z,,,,_l,,2,,

(y!?;~~~+~

/G

QQex~(--D4N4p),

x = 0, f l,...) +t(x,+, - 1); s = 0, l)...) Se+1 - 1, D>D,>D,.

(32)

Now by virtue of (18), (21), (22), and (32) having repeated the argument of (31) we obtain I &,,(0/

D

>

641/9/1-s

D,,

x

=

0,

< I g,*,(‘d/

+ I &A4

f (s + l)O

&25Q(e--VD*N4’

fl,...,

f&l+1

-

1);

- gs.,(Ol +

e-VD’N),

s = 0, l,..., s9+1 - 1.

(33)

58

N.

I.

FELDMAN

The nonzero numbers g,,,(5) satisfy .the inequality virtue of (15), (25), (28), and (29)

(26), therefore by

s+lp+lC exp(Q(3,25 + ln(4 + 2D2m In m))) I g,,,(5)/ 2 MQ + 1) K+‘+~ x (D3m” + 2q)s~+l)1-y

3 exp{-v(ln(2v2D5N)

L-YS~+“n

+ 2D3N In1 m ln(vD*)

+ (4,25 + In 4) D3N + D3N In1 m * ln(2 + D2m In m) + D32pf1 In-l m ln(D3m2 + 203)).

For D 3 D, 3 D, inequalities

(33) and (34) are inconsistent, so that

x = 0, &I,...,

&c(5) = 0,

(34)

f (2X, - 1);

s = 0, l)...) sp,, - 1,

D 3&t

and the Fundamental Lemma is proved. For p = 0 the hypotheses of the Fundamental Lemma thus after successive applications of this Lemma we have

For D-3m-2q, inequalities

< 2D5Nv/(D3m2)

< Qoe-DsN ,

/ f(xD-3m-2)1

< 2D2N/m

then, in particular,

0

the

x = 0, I,..., qo, qo = qQ + q - 1, (35)

take place. Using Lemma 9 and the equality Pk(xD-3m-2) find the quantities C,,, from the system f(xD-“rn-“)

are satisfied,

= R,(x) we can

B-l

= 1 1 Ck,&(x) k=O I=0

py,

p1 = exp(2rriD-3m-2),

a, = plz.

(36) We obtain aO+a-1 Ck3

=

2 I==0

(~k.d~)fW3

m-"1

0 o+a-1

= (l/Q!)

c ES?=0

O-k

f(xDe3me2)

c

(-l)“+s+k

p~“-““Ck,-,

a=0

Q-S

x 1 (l/r! x!) N,$)&O) . (M;l(y))“l ?I D,l. C==O

(37)

APPROXIMATION

OF

59

rr

We have

(l/x!)

I N~$.,,(O)I < c;,;-,c;;,

(W(Y))“’

= (y-0

5

n c;+l Pfl

< 23ao+4a,

(38)

(y - plp)-o-l)“’

(E-1)'r! Q
-

<

22r+a+aO(D3m2)a0-0+a-l+r

<

(D3m2)aO+9

plP)o+l+T,

'

290+4+27,

(39

because q -=c D3m2 and for p # I / pll - pip I > I 1 - p1 I = sin(,rr/D3m2) > 2/D3m2 > D-3m-2.

Now from (15), (35), (36)-(39) we obtain

I G,z I < (qQ + d g <

e-‘@N(Q

+

1)2

2309+0+4a(D3m2)aO+a

290+2O+a

e-v@N,

D > D, 3 D, , k = 0, l,..., Q;

The quantities C,,, are polynomials or by (15) and (16) we have

I c,,, I = j “($;l c*JJp7 1 2 yJ (Lrn)=l

I = 0, l,..., q -

1. (40)

in p, so either they are equal to zero

/ ‘($;l C?&p~t 1-l > v-y2-yDSN. (41)

60

N. I. FELDMAN

The inequalities

(40) and (41) are inconsistent, hence

G,, = 0,

k = 0, I,..., Q; I= 0, l,...,q - 1; D > D,.

But p is an algebraic number of degree y(m) hence all the Ck,1,T = 0 and that contradicts to our choice of them. Thus for D 2 D, the inequality (14) is impossible. We have only to set c,, = 2([D,] + 1)‘. REFERENCES

1. J. POPKEN, Zur Transzendenz von =, Math. Z. 29 (1929), 542-548. 2. C. L. SIEGEL, Uber einige Anwendungen Diophantische Approximationen, Abh. preuss. Akad. Wiss. 1 (1929-1930), I-70. 3. K. MAHLER, Zur Approximation der Exponentialfunktion und des Logarithmus, II, J. Reine Angew. Math. 166 (1932), 137-150. 4. K. MAHLER, On the approximation of r, Proc. Koninkl. Nederl. Akad. Wet. A 56 (1953), 30-42. 5.

6. 7. 8. 9. lo.

11. 12. 13.

H.

Ef.

~eJIS,JQMaH,

hIIpOKCHM3IJSiR

HeKOTOpbIX

TpaHCL(eHaeHTHbIX

WSCeJI

I,

ElssenHff AH CCCP, cepwn MareM. 15 (1951), 53-74; Amer. Math. Sot. Trunsl. Ser. 2 59 (1966), 224-245. H. II. 0enbnMan, 0 Mepe rpancuerrnenrnocrn gncna n, IIasecrrm AH CCCP, cepnn MareM. 24 (1960), 357-368. Amer. Math. Sot. Transl. Ser. 2, 58 (1966), 110-124. H. II. 0enbnMan, 0 npn6nmxermu nucna n anre6pau~ecrorMn WI~~~MII noneii, 06pa30B3HHbIX KOPHRMI~ 113 e@HtiqbI, TpynbI MOCK. reonoropa3B. IlHcTnryra 36 (1959), 188-198. K. MAHLER, An application of Jensen’s formula to polynomials, Mathematika 7 (1960), 98-100. J. B. ROF,ER AND L. SCHOENFELD,Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 6494. H. II. @eJIbAMaH, YJIysIIIeIGie OIJeHKIi JIPiHetiHOii +OpMbI, OT JIOrap&i@MOB anre6pamrecmix sxen. MareM. c6. 77 (1968), 423-436. Math. USSR Sb. 6 (1968), 393406. E. LANDAU, “Handbuch der Lehre von der Verteilung der Primzahlen,” Bd 1, Leipzig, 1909. B. JI. Iomrapon, Teoparr mi-repnormposamm M rrpu6rmwemm ~y~muiii, MOCKsa, 1934. E. WIRSING, Approximation mit algebraischen Zahlen beschrankten Grades, J. Reine Angew. Math. 206 (1961), 67-77.