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 13 Proof. It is known that the distance from the point (ZQ , yo, to the line L is , , , _ {^/DXQ - Zp) + {axp - yo)

2(VDyo - az

oy (5; — \ / 1 + a

2 + D By (1), (2) and (5), we have 1 + a

2+ű (6) U0">+0**{=£*f-)

2 + 0

2*{-0-°)

2 _ /

/3

2"(5 + A

2

) _ \ oi

n / 5+ A

2 ' - Y 1 + a

2 + D ~ V l + «

2 + D ~ y 1 + a

2+D " Prom this the theorem follows. It is easy to see that points P

n are on a plane. We investigate whether there is a lattice point P — (x,y,z) in the plane such that jxj < |Ä

n| and P is nearer to the line L than P

n. We use the previous denotations. Theorem 2. The points P

n = (R

n

, i?

n+i

, V

n) are in a plane. Furthermore if n is sufficiantly large, than there is no lattice pont (x,y,z) in this plane such that d

XiVt

Z < d

n and |x| < |Ä

n| if and only if \B\ — 1. Proof. First suppose |j9| = 1. In this case, obviously, \ß\ < 1 and a is irrational, as it was supposed. Using (2), we have £

n

+i = + = + /T a

n - ß

n aß

n - ß

n+l a — ß a — ß and similarly R

n+

1 = ßR

n + OL

n

. Adding these equation, we get (7) 2 R

n+

l =(a + ß)R

n + V

n. Consequently, the points P

n are on the plane which is defined by the equation Ax — 2y + z = 0. It is easy to prove that L is also on this plane. Assume that for some n there is lattice point (x,y,z) on this plane such that (8) d

X y Z 5Í d

n