Approximation to real numbers by algebraic integers

Exponents of Diophantine Approximation and Sturmian Continued Fractions

“…For a positive real number x, we denote by x the smallest integer ≥ x. The following statement translates the main results of [14] and of [22] in terms of exponents: THEOREM 2.5. -For any integer n ≥ 1 and any transcendental real number ξ, we have…”

“…Some results of [14] have been recently improved by Laurent [22]. For a positive real number x, we denote by x the smallest integer ≥ x.…”

“…Davenport & Schmidt [14] have investigated the approximation to a real number by algebraic integers, using a dual approach which may also be applied to Wirsing's Conjecture. Indeed, they established the inequality w * n (ξ) ≥ w n (ξ) for any positive integer n and any real number ξ, which links up approximation to ξ by algebraic numbers of degree at most n with uniform simultaneous rational approximation to ξ, .…”

“…In particular, Davenport & Schmidt [14] have investigated this question in a dual way, via simultaneous rational approximation to the successive powers of ξ. Among other results, they proved that if the real number ξ is either transcendental, or algebraic of degree ≥ 3, then there exist some positive constant c, depending only upon ξ, and arbitrarily large real numbers X such that the inequalities (2) 0 < |x 0 | < X,…”

Sur une question de N. Chevallier liée à l’approximation diophantienne simultanée

“…Le premier résultat remonte à V. Jarník [6] (voir aussi [3] et [7]). Il affirme que pour n ≥ 2 il y a un nombre infini de triplets de meilleures approximations consécutives (g ν , g ν+1 , g ν+2 ) formés de vecteurs linéairement indépendants.…”