On the multiples of a badly approximable vector

Yann Bugeaud¹

¹ Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg Cedex, France

Acta Arithmetica, Tome 168 (2015) no. 1, pp. 71-81

Résumé

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 $n \ge 1$.

Détail

DOI : 10.4064/aa168-1-4

Keywords: positive integer alpha real algebraic number degree set underline alpha alpha alpha ldots alpha well known underline alpha mathop minf infty cdot underline alpha where cdot denotes distance nearest integer furthermore underline alpha underline alpha underline alpha integer main result asserts there exists real number nbsp nbsp depending only alpha underline alpha integer