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.

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