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On the multiples of a badly approximable vector

Volume 168 / 2015

Yann Bugeaud Acta Arithmetica 168 (2015), 71-81 MSC: Primary 11J13. DOI: 10.4064/aa168-1-4

Abstract

Let $d$ be a positive integer and $\alpha$ a real algebraic number of degree $d+1$. Set $\underline\alpha := (\alpha, \alpha^2, \ldots , \alpha^d)$. It is well-known that $$ c(\underline\alpha) := \mathop{\rm li}minf_{q \to \infty} \, q^{1/d} \cdot \| q \underline\alpha \|>0, $$ where $\| \cdot \|$ denotes the distance to the nearest integer. Furthermore, $$ {c(\underline\alpha) n^{-1/d}} \le c(n \underline\alpha) \le n c(\underline\alpha) $$ for any integer $n \ge 1$. Our main result asserts that there exists a real number  $C$, depending only on $\alpha$, such that $$ c(n \underline\alpha ) \le C n^{-1/d} $$ for any integer $n \ge 1$.

Authors

  • Yann BugeaudUniversité de Strasbourg
    Mathématiques
    7, rue René Descartes
    67084 Strasbourg Cedex, France
    e-mail

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