About fifty years ago Mahler proved that *if
is rational but not an integer and if , then the fractional
part of is apart from a finite set of integers depending
on and *. Answering completely a question of Mahler, we show that
the same conclusion holds for all algebraic numbers which are not -th roots of
Pisot numbers. By related methods we also answer a question of
Mendès France, characterizing completely the real quadratic irrationals
such that the continued fraction for has period length tending to infinity.

Please send comments and corrections to Thomas Klausner.