Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

DRESS, A. & LUCA, L., Real numbers that have good diophantine approximations

Real numbers that have good diophantine approximations 17 holds for all n > no- Prom inequalities (7) for n and n 1 and the triangular inequality, we get IM+i - r

nr

n+

2

1 _ rn I'n

+1 'n

+1 < ?n + l + r

n+ 2 < a + a r

n ?'n+l \ / 1 _1 v + ,!rn 7 »1 -I 71 + 1 (14) r

n+

1 - r

n+

2r

n\ ^ r' y/Tn+l Using inequality (13) in (14), we get 71 rn+1 1 \l + l V?

>71 + 1 (is; 71 + 1 r

n+2r

n\ ^ ci c

2 \An + l 7 7' 72 + 1 for all 7? > ?io, where C\ = and c

2 = 1 /S. We now let n tend to infinity in (15) and get (10) lim n—*oo r

n+

1r

n_i| _ o 1 yft' Consequently, the conclusion of part (i) of Theorem 1 follows from part (i) of Theorem DL. The remaining assertions of part (ii) now follow from putting e := 1/2 in (15) and invoking r

n+i/r

n < 7 as well as part (ii) of Theorem DL. Theorem 1 is therefore established. Remark 1. The occurence of e > 0 in the exponent in inequality (7) is unnecessary. A closer investigation of the arguments used in the proof of Theorem 1 shows that the conclusion of part (i) of Theorem 1 remains valid if inequality (7) is replaced by the weaker inequality (7': 71 + 1 < 1 - € 1 v^x/M + i/H Remark 2. Assume that cv is a real number such that the hypotheses of either part (i9 or part (ii) of Theorem 1 are fulfilled. Using the full strength of our results from [1] and [2], we can infer that if a is an integer, then (r

n)

n>o is a geometrical progression of ratio cv from some n on. However, if a is quadratic and the hypotheses of part (ii) of Theorem 1 are fulfilled, we can only infer that (7'

n)7i>o is binary recurrent from some n on, and that its charateristic equation is precisely the minimal polynomial of a over Q. However, we cannot infer that (r

n)

n>o is the