Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

LlPTAI, K., An approximation problem concerning linear recurrences

An approximation problem concerning linear recurrences KÁLMÁN LIPTAI* Abstract. Let {/in}T=o an

c^ {^nj-^o ("=0,1,2,...) be sequences of integers defined by R

n=AR

n-i-BR

n2 and V

n =AV

n_

1-SV„_

2, where A and B are fixed non-zero integers. We prove that the distence from the points P

n(R

n ,R

n + 1 ,V

n) to the line L, L is defined by x=t,y=at,z—V~Dt, tends to zero in some case. Moreover, we show that there is no lattice points (x,y,z) nearer to L than P

n(R

n ,R

n

+ i,V

n) if and only if |ß|=l. Let {R

N}%L

0 and {V

n}^_

0 be second order linear recurring sequences of integers defined by R

N = AÄ

n-i - BR

N-2 (n > 1), y

n = Ay

n_! - BV

N.2 (n > 1), where A > 0 and B are fixed non-zero integers and the initial terms of the sequences are Rq = 0. R\ = 1. Vq = 2 and V\ = A. Let a and ß be the roots of the characteristic polynomial x

2 — Ax + B of these sequences and denote by D its discriminant. Then we have (1) VD = \JA

2 -4 B = a-ß, A = a + ß

, B = aß. Throughout the paper we suppose that D > 0 and D is not a perfect square. In this case, a and ß are two irrational real numbers and |a| ^ \ß\, so we can suppose that J or | > \ß\. Furthermore, as it is well known, the terms of the sequences R and V are given by a

n - 3

n (2) R

N= and V

n = a

n + ß

n

. a — p Some results are known about points whose coordinate are terms of linear recurrences from a geometric points of view. G. E. Bergum [1] and A. F. * Research supported by the Hungarian National Scientific Research Foundation, Operating Grant Number OTKA T 016975 and 020295.