On the multiples of a badly approximable vector

Yann Bugeaud. "On the multiples of a badly approximable vector." Acta Arithmetica 168.1 (2015): 71-81. <http://eudml.org/doc/286164>.

@article{YannBugeaud2015,
abstract = {Let d be a positive integer and α a real algebraic number of degree d + 1. Set $α̲:= (α,α²,...,α^\{d\})$. It is well-known that $c(α̲) := lim inf_\{q→ ∞\} q^\{1/d\}·||qα̲|| > 0$, where ||·|| denotes the distance to the nearest integer. Furthermore, $c(α̲)n^\{-1/d\} ≤ c(nα̲) ≤ nc(α̲)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c(nα̲) ≤ Cn^\{-1/d\}$ for any integer n ≥ 1.},
author = {Yann Bugeaud},
journal = {Acta Arithmetica},
keywords = {Lagrange constant; badly approximable vector; simultaneous approximation},
language = {eng},
number = {1},
pages = {71-81},
title = {On the multiples of a badly approximable vector},
url = {http://eudml.org/doc/286164},
volume = {168},
year = {2015},
}

TY - JOUR
AU - Yann Bugeaud
TI - On the multiples of a badly approximable vector
JO - Acta Arithmetica
PY - 2015
VL - 168
IS - 1
SP - 71
EP - 81
AB - Let d be a positive integer and α a real algebraic number of degree d + 1. Set $α̲:= (α,α²,...,α^{d})$. It is well-known that $c(α̲) := lim inf_{q→ ∞} q^{1/d}·||qα̲|| > 0$, where ||·|| denotes the distance to the nearest integer. Furthermore, $c(α̲)n^{-1/d} ≤ c(nα̲) ≤ nc(α̲)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c(nα̲) ≤ Cn^{-1/d}$ for any integer n ≥ 1.
LA - eng
KW - Lagrange constant; badly approximable vector; simultaneous approximation
UR - http://eudml.org/doc/286164
ER -