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

Acta Acad. Paed. Agriensis, Sectio Mathern,aticae 28 (2001) 13-19 REAL NUMBERS THAT HAVE GOOD DIOPH

ANTINE APPROXIMATIONS OF THE FORM r

n+

1/r

n Andreas Dress & Florian Luca (Bielefeld & Morelia) Abstract. In this note, we show that if a is a real number such that there exist a constant C and a sequence of non-zero integers (?*n)n>0 with lim

n_

+c

o |?'

n| — OO for which — • "

< -—— holds for all 71 > 0, then either a E Z\{0, ±1} or a is a quadratic J'n 11'n I

" unit. Our result complements results obtained by P. Kiss who established the converse in Period. Math. Hungar. 11 (1980), 281-187. AMS Classification Number: 11J04, 11J70 1. Introduction Let a be a real number. In this paper, we deal with the topic of approximating a by rationals. It is well known that there exist a constant c and two sequences of integers (u

n)

n>o and (i>

n)n>o with v

n > 0 for all n > 0 and v

n diverging to infinity (with n

) such that (1) u, v

, c < — vi holds for all n > 0. By work of Hurwitz (see [5]), one can take c := l/\/5 and the 1 "n/Ö above constant is well known to be best-possible for a := —-—. Several papers in the literature deal with the question of approximating cv by rationals u

n/v

n requiring u

n and v

n to satisfy (1) as well as some additional conditions. For example, if a is irrational and a, b and k are integers with k > 1, then there exist a constant c and two sequences of integers (tt

n)n>o a

i*d (v

n)

n>o with v

n > 0 and v

n diverging to infinity such that (2) u, a Vr c < — and u

n = a (mod k

), v

n = b (mod k) vThe second author's research was partially sponsored by the Alexander von Humboldt Foundation.