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

44 Béla Zay It is easy to see that e

C

l < H\ (n) < e°

2 holds with suitable real numbers ci, c

2. Prom this it follows that (5) ce

hn+

c> < cH

x(n)H

2(n) < ee

hn+a

a where h = log a

2 + (T* — 1) log c*i

. If we assume that |a

2| > l

a3| then lim cHi{n)Hz(n) = cc

0, where n—* oo c

0 = |oif

1a

2i(af - a|)|. Using the well known fact [0, if r < ro = 1 - I^S lim H

2(n) = lim aiaj = < i if r = r

0 n—»-oo n—••oo I I oo, if r > r

0 it is clear that (4) (and so (2), too) holds for infinitely many positive integers n with some positive constant c (0 < c < Cq

1) if and only if r < r

0. Now we assume that M > Kl = |«

3| > ja

4j > > |a

t| . Since ai is real and az/a.

2 is not a root of unity a

3 and a

2 are (not real) conjugate complex numbers and m

2 = (i.e. m\ — ra

2 = 1 = rnz and P

3(n) = P

3(n + 5) = a

3i). Furthermore a

2

1 a3i also are conjugate numbers since they are solutions of the system of linear equations t (mi Gn = i 0 < n < k - I. i=

1 \i=l Hence a

3'

1i°

!

l~^

! and ^

4 are algebraic numbers with absolute value 1 «2,1(0^-a*) a

2 0 and so using the Lemma (proved by P. Kiss in [1]), we obtain the estimation l + a

3i(af - a|) f a

3 a

2i(af - a|) \a

2 with some positive real 6. y log n