Simultaneous rational approximation to the successive powers of a real number

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On Exponents of Homogeneous and Inhomogeneous Diophantine Approximation (2005, background): “The first upper bound is the main result of [37], while Theorem 2b of [12] is equivalent to the second one.”
Exponents of Diophantine approximation (2006, background): “Shortly thereafter, Bugeaud & Laurent [11] extended Roy’s construction and found uncountably many values taken by ŵ2 and λ̂2.”
Approximation to Real Numbers by Cubic Algebraic Integers I (2002, background): “where the upper bound comes from standard metrical considerations, while the lower bound, due to M. Laurent [ 4 ], refines, for even integers n, the preceding lower bound τn ≥� (n +1 )/2� of Davenport and Schmidt [3].”
On transfer inequalities in Diophantine approximation (2007, background): “See the articles [5, 13, 18, 19] and the reports [2, 20] for relating those exponents with various results on algebraic approximation.”