Pietro Corvaja: Rational approximations to the powers of an algebraic number

This is a joint work with Umberto Zannier.

About fifty years ago Mahler proved that if $ \alpha>1$ is rational but not an integer and if $ 0<l<1$, then the fractional part of $ \alpha^n$ is $ >l^n$ apart from a finite set of integers $ n$ depending on $ \alpha$ and $ l$. Answering completely a question of Mahler, we show that the same conclusion holds for all algebraic numbers which are not $ d$-th roots of Pisot numbers. By related methods we also answer a question of Mendès France, characterizing completely the real quadratic irrationals $ \alpha$ such that the continued fraction for $ \alpha^n$ has period length tending to infinity.

Back to the Index

Please send comments and corrections to Thomas Klausner.