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 15 Theorem 1. Let a be a real number. (i) Assume that there exist e > 0 and a sequence of integers (r

n)

n>o with \r

n diverging to infinity such that (7) n+l < 1 holds for all n > 0. Then, a is a real algebraic integer of absolute value larger than 1 and of degree at most 2. Moreover, if a is irrational, then the absolute value of its norm is smaller than \/jck[. (ii) Assume, moreover, that there exist a constant c and a sequence of integers (

rn

)n >

o with |r

n| diverging to infinity such that (8) n + l < holds for all n > 0. Then a is a quadratic unit or a rational integer different from 0 or ±1. The following result characterizes real numbers a for which - as in (5) - two different powers can be well approximated by rationals. Theorem 2. Let a be a real number. Assume that there exist two coprime positive integers si and so, two positive integers t\ and 1

2, a real number e > 0, and a sequence of integers (r

n)

n>o with |r

n \ diverging to inßnity with n such that (9) I'n+t, < 1 hold for all n > 0 and for both i — 1 and 2. Then, either a £ Z\{0, ±1} or a is quadratic irrational with norm smaller than \/|cvj in absolute value. If moreover a is irrational and there exists a constant c with (10) n+i, C < — then a is a quadratic unit. The proofs of both Theorems 1 and 2 are based 011 the following result which follows right away from our recent work [1] and [2], Theorem DL. Let (r

n)

n> 0 be a sequence of integers with |r

n| diverging to inßnity. (i) Assume that (ID lim. 1 r

n — rn+lT.n-l

l 1 71'