A problem of Mahler on farctional parts of powers of an algebraic number is solved, namely a classification is provided of the algebraic numbers $\alpha$ such that the fractional powers of $\alpha^n$ tends to zero exponentially on a sequence of integers. A problem of Mendes France is solved, by proving that the period length of the continued fraction of the powers of a quadratic irrational tends to infinity apart trivial cases.

On the rational approximations to the powers of an algebraic number. Solution of two problems by Mahler and Mendes France

Abstract

A problem of Mahler on farctional parts of powers of an algebraic number is solved, namely a classification is provided of the algebraic numbers $\alpha$ such that the fractional powers of $\alpha^n$ tends to zero exponentially on a sequence of integers. A problem of Mendes France is solved, by proving that the period length of the continued fraction of the powers of a quadratic irrational tends to infinity apart trivial cases.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/852245