On the multiples of a badly approximable vector

Yann Bugeaud

Yann Bugeaud

Acta Arithmetica, Tome 168 (2015), p. 71-81 / Harvested from The Polish Digital Mathematics Library

Résumé

Let d be a positive integer and α a real algebraic number of degree d + 1. Set $α ̲ : = (α, α ², . . ., α^{d})$ . It is well-known that $c (α ̲) : = l i m i n f_{q \to \infty} q^{1 / d} \cdot | | q α ̲ | | > 0$ , where ||·|| denotes the distance to the nearest integer. Furthermore, $c (α ̲) n^{- 1 / d} \leq c (n α ̲) \leq n c (α ̲)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c (n α ̲) \leq C n^{- 1 / d}$ for any integer n ≥ 1.

Publié le : 2015-01-01

Zbl 1325.11062

EUDML-ID : urn:eudml:doc:286164

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     title = {On the multiples of a badly approximable vector},
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     volume = {168},
     year = {2015},
     pages = {71-81},
     zbl = {1325.11062},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-1-4}
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Yann Bugeaud. On the multiples of a badly approximable vector. Acta Arithmetica, Tome 168 (2015) pp. 71-81. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-1-4/