Some metrical observations on the approximation by continued fractions

Approximation by mediants

“…The main tool we will use is a variation on a theme that first appeared in [1] and was used in several papers thereafter. The theme consists of considering the sequence {(Tn(x), ^"(*))}^10 for an irrational number x , where Tn(x) is given by G\r>(z) = o,…”

Metrical properties of best approximants

“…This is expressed more precisely by the fact that the inequality & n < 1 always holds. One also has the following theorem which was conjectured by H. W. Lenstra and proved in [3]; see also [5] and [6]. …”

“…Now we will do some preparatory work for Theorem (2.16) and Theorem (2.18). We note that we will use methods from [2] which in their turn go back to [3]. All the metrical results we will find are based on ergodic theory by using the following theorem.…”

On the Lenstra constant associated to the Rosen continued fractions

“…We denote by γ (x, y) the geodesic curve with initial point x and terminal point y for (x, β) ∈ (R ∪ {∞}) 2 \ {diagonal}. We also denote by…”

“…Legendre and Lenstra constants can be defined for other types of continued fraction expansions in a similar manner (e.g. for α-expansions, [16], [2]). In general, it is not hard to show that the Lenstra constant exists and is at least as large as the Legendre constant for each type of continued fraction expansion whenever the Legendre constant exists.…”