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

16 A. Dress fe F. Luca Then, the sequence ( —— ) is convergent to a limit a that is a non-zero \ r

n / n>o algebraic integer of degree at most 2. If a is irrational, then its norm is smaller than y/\a\. Moreover, there exists no £ N such that (r

n)

n>

n

o is binary recurrent. (ii) If (12) l

rn ~ rn+lTn-l| < C holds for some constant c and all n, then a is a quadratic unit or a non-zero integer. We point out that in our work [1] and [2], we gave more precise descriptions for both the sequences (r

n)

n>o satisfying (11) or (12), respectively, and the limit a = lim 71+

1

, but the above Theorem DL suffices for our present purposes. n — oo r

n We now proceed to the proofs of Theorems 1 and 2. 2. The Proofs Proof of Theorem 1. We will prove (i) in detail and we will only sketch the proof of (ii) . (i) By replacing the sequence (r

n)

n by the sequence ((—l)

nr„) and a by —a if a < 0, we may assume a > 0 and r

n > 0 for all n > 0. By letting n tend to infinity in (7), we get a — lim n+

1

. Since r

n diverges to infinity, we must have n —+ co V

n a > 1. We now show that a > 1. Indeed, if a — 1, then inequality (7) becomes 1 1 _ r»+

1 < Vn or 1 \

rn+l - rn I < < !> ?n therefore r

7J

+i = r

n for all n > 0. This contradicts the fact that r

n diverges to infinity. Hence, a > 1. Now let 6 be a real number with 1 < 8 < a, note that j := 2a — 6 exceeds a, and choose no such that ^ 1 Vn > a — 6 holds for all n > no- From inequality (7), we get that (13) Si