Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

ZAY, B., A generalization of an approximation problem concerning linear recurrences

A generalization of an approximation problem concerning linear recurrences 45 But |a!j| < |c*21 for i > 4, so by the last inequality e~C

3 log 71 ^ (6) 1 + Q3(

Q1 - a3

) ( 7

1 +£ i=

4 af P{(n) - a?Pí(n + s) ( a a

2i(af - a|) «2 = ^s(n) < 3 with some C3 > 0 if n is large enough. By (5) and (6) we have (7) C

ehn-c, logn+cx < c^ (

n)//

2 (

n) < CgAn+c

2+log 3 _ (7) holds for infinitely many positive integers if and only if h < 0, which is equivalent to r < r

0. This completes the proof of the theorem. References [1] P. KLSS, An approximation problem concerning linear recurrences, to appear. [2] P. KLSS, A Diophantine approximative property of the second order linear recurrences, Period. Math. Hungar. 11 (1980), 281-287. [3] P. Kiss AND ZS. SINKA, On the ratios of the terms of second order linear recurrences, Period. Math. Hungar. 23 (1991), 139-143. [4] P. KLSS AND R. F. TLCHY, A discrepancy problem with applications to linear recurrences I., Proc. Japan Acad. 65 (ser A), No 5. (1989), 135-138. [5] P. KLSS AND R. F. TLCHY, A discrepancy problem with applications to linear recurrences II., Proc. Japan Acad. 65 (ser A), No 5. (1989), 131-194. BÉLA ZAY ESZTERHÁZY KÁROLY TEACHERS' TRAINING COLLEGE DEPARTMENT OF MATHEMATICS LEÁNYKA U. 4. 3301 EGER, PF. 43. HUNGARY