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

14 A. Dress fe F. Luca holds for all n > 0. The best-possible constant c in (2) Is k

2 /

4 in case a and b are not both divisible by k (see [3] and [4]). If a is algebraic and V is a fixed finite set of prime numbers, then Ridout [10] inferred from Roth's work [11] that one cannot approximate a too well by rational numbers u/v where either u or v is divisible only by primes from V

. More precisely, for every given e > 0, the inequality (3) < ,1 + C has only finitely many integer solutions (u, v) with v > 0 and either u or v divisible by primes from V, only. A different type of question was considered by P. Kiss in [6] and [7] (see also [8] and [9]). In [6], it was shown that if cv is a quadratic unit with |a| > 1, then there exist a constant c and a sequence of integers (r

n)

n>o with |r

n| diverging to infinity such that (4) "n + l < holds for all n > 0. In [7] it was shown that, in fact, a statement similar to (4) holds for both a and a

s where s > 2 is some positive integer: There exist a constant c and a sequence of integers (r

n)

n>o with \r

n\ diverging to infinity such that both 15) n + l < and < hold for all n > 0. An explicit description of a sequence (r

n)

n>o satisfying inequalities (5) above was also given in [7]: Let f = X

2 + AX + B (A, Be Z) be the minimal polynomial of a over Q. Let ß be the other root of /. Since a is a unit, \B\ — \cxß\ = 1 must hold which implies that the sequence (6) ß

T <x-ß ' n > 1 fulfills the inequalities (5) for all n with c := 2^0 MÍ/?|

s~

1-?

:One may ask if one can characterize all real numbers a for which there exist a constant c and a sequence of integers (r

n)

n>o with |r

n| diverging to infinity such that inequality (4) or, respectively, inequalities (5) hold for all n > 0. Fi'om the above remarks, we saw that quadratic units cv with |a| > 1 have these properties. Moreover, the sequence r

n :— a

n (n > 1) shows that integers a with |a| > 1 also belong to this class. It seems natural therefore to inquire if there are any other candidates cv satisfying the above conditions. The perhaps not too surprising, answer is no. Our exact result is the following.