On badly approximable numbers

Abstract

Motivated by a wonderful paper [7] where a powerful method was introduced, we prove a criterion for a vector α α α ∈ R d to be a badly approximable vector. Moreover we construct certain examples which show that a more general version of our criterion is not valid.

A note on Dirichlet spectrum

“…In particular Theorem 1 together with the result from [2] show that for any λ ∈ D [2] |•| there exists θ θ θ ∈ R 2 which is not badly approximable and d…”

“…Of course the sequence (2) of the best approximations depends on the norm g. However as all the norms in R n are equivalent, from Theorem 1 from the paper [2] we know that θ θ θ ∈ R n is badly approximable if and only if the inequality sup…”

“…We would like to mention that very recently a series of results concerning spectrum D [2] g for arbitrary norm g in R 2 was obtained in [19]. In particular it was proven there that the maximal point max D [2] g of the spectrum D [2] g is not isolated in D [2] g .…”

“…In the case n 2 not much is known, however a complete structure of the spectrum D in R 2 was discovered by Akhunzhanov and Shatskov in [1]. It turned out that D [2] |•| is a segment, namely…”

“…In particular it was proven there that the maximal point max D [2] g of the spectrum D [2] g is not isolated in D [2] g . In this paper we are interested in a more detailed analysis of distribution of the values d [2] g (θ θ θ) = d [2] g (θ 1 , θ 2 ) in the two-dimensional case θ θ θ = (θ 1 , θ 2 ) ∈ R 2 .…”