Cubic approximation to Sturmian continued fractions

Abstract

We determine the classical exponents of approximation w 3 (ζ), w * 3 (ζ), λ 3 (ζ) and w 3 (ζ), w * 3 (ζ), λ 3 (ζ) associated to real numbers ζ whose continued fraction expansions are given by a Sturmian word. We more generally provide a description of the combined graph of the parametric successive minima functions defined by Schmidt and Summerer in dimension three for such Sturmian continued fractions. This both complements similar results due to Bugeaud and Laurent concerning the two-dimensional exponents and generalizes a recent result of the author. As a side result we obtain new information on the spectra of the above exponents. Moreover, we provide some information on the exponents λ n (ζ) for a Sturmian continued fraction ζ and arbitrary n.

Diophantine Approximation in Prescribed Degree

“…Furthermore any Sturmian continued fraction defined as in [11] also provides a counterexample for certain indices. Indeed, for these numbers we have both w =2 (ζ) > w =3 (ζ) and w =2 (ζ) > w =3 (ζ), as follows from the results in [39]. See also [38,…”

Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type

“…To conclude it suffices to note that for each 0 < l < s k+1 , we have y ψ(t k +l) = y t k +l−1 = w l k y ψ(t k ) . Also note that (4.4) is equivalent to (19) of [5]. We find the first relation of (4.6) by using (4.4) together with the definition of z t k +l+1 and z t k +l .…”

“…(one may compare this estimate with (19) of [14]). Indeed, by (3.2), by multiplicative growth and with c 1 , c 2 > 0 denoting the constants of (5.1), we have…”

On Wirsing's problem in small exact degree

“…The bound appears to be optimal, extremal numbers ξ defined by Roy [20] satisfy [23]. Similar results apply for every Sturmian continued fraction [24]. Theorem 1.3 was known for n = 2 in view of ( 5) and (9).…”