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
42 Béla Zay where the coefficients aij of polinomials P;(n) are elements of the algebraic number field <5(QI, • • •, We assume that the sequence G is a non degenerate one, i.e. a\\ , a 2i,..., a t l are non zero algebraic numbers and ai/aj is not a root of unity for any 1 < i < j < t. We can also assume that G n / 0 for n > 0 since the sequence have only finitely many zero terms and after a movement of indices this condition will be fulfilled. If |ai| < ai for i — 2, 3,..., t than from (1) it follows that lim r" + 1 = a\ . In the case Jl— • OO k = 2 the quality of the approximation of a\ by rational numbers G n +i /G n was investigated some earlier papers (e.g. set [2], [*3], [4] and [5]). In the general case P. Kiss ([1]) proved the following result. Let G be a t t h order linear recurrence with conditions |ai| > |a 2| > jct 31 > ••• > |at|, where mi = ••• = m t = I). Then G n+ 1 ai G, < cGt holds for infinitely many positive integers n with some constant c if and only if k < where io = 1_^4< 1 + 1 log | a i| t — 1 and the equation ko = 1+ jb[ ca n t> e held only if \At\ = 1 and |o;i | > |qt2I = • •• = |OÍ|. In [1] the following lemma was also proved. Lemma. Let ß and 7 be complex algebraic numbers for which \ß\ — I7I = 1 and 7 is not a root of unity. Then there are positive numbers S and no depending only on ß and 7 such that \l + ßl n\ > e Älog 7 1 for any n > no. In the case |ai| > (2 < i < t ) it is clear that lim — a{ for n—+00 n any fixed positive integer s. The purpose of this paper is the investigation of the quality of the G approximation of a( by rational numbers and to prove an extension of P. Kiss's theorem. Theorem. Let G be a non degenerate k t h order linear recurrence sequence with conditions: t l öi I > 1^2 1 > l^l > I04 j > • • • > j, mi = m 2 = 1, y ^mj — k »=1
