K.C. PRASAD Department M. · K.C. PRASAD Department of Mathematics Ranchi University Ranchi 834008, India M. LARI Birsa Agriculture University P. SINGH S.S.M. College Ranchi, India

Internat. J. Math. & Math. Sci.VOL. 13 NO. (1990) 201-204

201

LOCATION OF APPROXIMATIONS OF A MARKOFF THEOREM

K.C. PRASAD

Department of MathematicsRanchi University

Ranchi 834008, India

M. LARI

Birsa Agriculture University

P. SINGH

S.S.M. CollegeRanchi, India

(Received November 26, 1986)

ABSTRACT. Relative to the first two theorems of the well known Markoff Chain (J.W.S.

Cassels, "An introduction to diophantine approximation" approximations are well located.

Literature is silent on the question of location of approximations in reference to the

other theorems of the Chain. Here we settle it for the third theorem of the Chain.

KEY WORDS AND PHRASES. Continued fractions, rational approximation.

1980 AMS SUBJECT CLASSIFICATION CODE. IOF05

I. INTRODUCTION

Suppose 8 is an irrational number whose simple continued fraction expansion is

[a0, aI, a2 a ]. Let e (8) denote

n n

0 a an-

a + an+ an+2n’ I’

Markoff Chain (Cassels [i], Kokshma [2]) is the following chain of theorems about the

sequence {=n(8)} n >=

TI: For every irrational number 8,

.(o) >/

for infinity of j’s and cannot be increased for 8 [0, (i)* ].

T2: If 8 [0, (i)*], then

for infinity of js and Vr cannot be increased for e [0, (2)*].

(1.1)

(1.2)

202 K.C. PRASAD, M. LARI AND P. SINGH

T3: If 8 [0, (I) or [0, (2) then

221 )/5

for infinity of j’s and / 221)/5 cannot be increased for 8 [0, (2,2,1,1)*])* * ,

T4: If 8 [0 (I or [0, (2) or [0, (2,2,1,1) then

=.(S) > ,/ 1517)/13

for infinity and j’s and / 1517)/13 cannot be increased for

(1.3)

(1.4)

0 [0, (2,2,1,1,1,1) etc.

It is know that the sequence of constants , , 21)/5, 517)/13increases to 3. So the theorems say something non-trivial about O’s in which all

quotients are eventually or 2 only.

As regards (1.1) and (1.2) we have an ad-hoc idea of the j’s satisfying them. In

reference to T we know that one j must occur in {n, n+l, n+2} ngl. Relative to

9 then a j e {n n+l, n+2} These may be foundT2, we have a similar result if an+2in Wright [3] or Prasad and Lari [4].

But the literature is surprisingly silent on such results in reference to T3, T4,etc. In this article we announce one such result in reference to T

3in the following

theorem:

2. MAIN RESULTS

THEOREM. an+2 2 and an+3then .(8) (Vl)/5 for at least one j e {n, n+l, n+4}

REMARK. Our method gives a way to try for similar results on T4, T5, etc.

PROOF. Suppose 8 [a0, a an

], an+2 2 and an+3If an+ 3; an(8) > 3 and we are through.

If an+ I; n+l(8) > [0,2] + [2,2] 3 and we are through.

If an+ 2 and an+4 2 then n+l(0) [0,3] + [2,1,1] 3 and we are through.

For the left out 8’s: an+ 2 an+2 and an+3 an+4.To deal with them, we put

[0, an, an_ al],[0, an+5, an+6 ],

t (,/1)/5

and argue over all possible values of 8 We note:

’n+4(0) t’ <- ’e[S-(5t-3)] > [(2t-7)B 12]’

SoB. 1212t-7 > n+4(8) t

LOCATION OF APPROXIMATION OF A MARKOFF THEOREM 203

We next check:

(8) > t<- a fl(8)(5t-12) + (3t-7)8

(5 + 38)

(12-4t) + (7-2t)8an+l(8) > t a f2(8)= (2t-5)

and f2(8) fl(8) A (8 + 25114- 5t) 85tI 9

where A1

t(10-3t) (5+38) -I {(2t-5) (3-t)8} -I (> 0).

5t-9So 8 >14

--> f2(8) fI(8)

a fl(8) or a < f2(8)

(8) > t or an+ (8) > t.

Hence we confine attention to12 5t-9

12t-7 8 &14"

In this case an+4(8) > t< a > f3(8) (12t-7)8 -125-(5t-3)8

5t+19 (8 5t-9Also f3(8) f2(8)= A2(8 + 14 -where A

22t(t-l) {5-(5t-3)8} -I {(2t-5) (3-t)8} -I (>0)

12 5t-9So if

12t-7< 8 <

14 then f3(8) f2(8)

which implies a < f2(8) or a > f3(8); equivalently

an+l (o) > t or an+4(8) > t and we are through.

5t-9Finally 8 14

5t-9=> fl(8) f2(8) f3(8) (an irrational number)

--> a fl(8) or a f2(8), a is rational)

(8) > t or an+l(8)_ > t.

This completes the proof of the theorem.

204 K.C. PRASAD, M. LARI AND P. SINGH

REFERENCES

i. CASSELS, J.W.S. An Introduction to Diophantine Approximation, CambridgeTracts No. 45, Cambridge, 1957.

2. KOKSMA, J.F. Diphatische Approximation, Chelsea, New York, 1936.

3. WRIGHT, E.M. Approximation of Irrationals by Rationals, Math. Gaz. 48 (1964),288-89.

4. PRASAD, K.C. and LARI, M. A Note on a Theorem of Perron; Proc. Amer. Math. Soc.97 (1986) 19-20.

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