An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation

Singular vectors on manifolds and fractals

“…), which is valid for all totally irrational ξ ξ ξ ∈ R n . This result was proven in [26], and a short and beautiful proof was given recently in [28].…”

On badly approximable numbers

“…Proof. In fact, this lemma follows from inequality (59) of Lemma 10 from [3]. For the sake of completeness we give here a proof.…”

“…As it was discovered by V. Jarník [11], the first trivial inequality here can be improved. The optimal lowed bound for ω(α α α) in terms of ω(α α α) was obtained in [3] where the authors solve a problem by W.M. Schmidt and L. Summerer [10].…”

“…We use a standard trick which reduces the problem for linear forms to the problem for simultaneous approximation. This trick was used in [3], Section 5.2.…”

An improved bound in Wirsing's problem

“…However, refinements in the constant term were made first by Bugeaud and Schleischitz [5]. The proof strategy in [5], in the light of later findings [19,12], in turn yields slightly stronger bounds, in particular w n (ξ) ≤ 2n − 2 for n ≥ 10. See also [21], where a conjectural bound of order (1 + 1/ √ 2)n − o(1) < 1.71n was motivated as well.…”