Geodesic
Partitioning of the interval $[0,1]$ induced
Sbornik. Mathematics, Tome 198 (2007) no. 5, pp. 661-690 
A. A. Dushistova. Partitioning of the interval $[0,1]$ induced. Sbornik. Mathematics, Tome 198 (2007) no. 5, pp. 661-690. http://geodesic.mathdoc.fr/item/SM_2007_198_5_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $p_{i,n}$, $i=1,\dots,2^{n-1}$, be the lengths of the intervals between successive points in the Brocot sequence $F_n$. An asymptotic formula for the quantity $\sigma(F_n)=\sum_{i=1}^{N(n)}p_{i,n}^\beta$, which improves the previously known estimates, is obtained. Bibliography: 5 titles.

[1] A. Brocot, Calcul des rouages par approximations, nouvelles méthodes, par Brocot, horloger (Paris), 1892

[2] E. Lucas, Théorie des nombres. Tome I. Le calcul des nombres entiers. Le calcul des nombres rationnels. La divisibilité arithmétique, Gauthiers–Villars, Paris, 1891 | Zbl

[3] R. Grekhem, D. Knut, O. Patashnik, Konkretnaya matematika. Osnovanie informatiki, Mir, M., 1998; R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics: a foundation for computer science, Addison-Wesley Publ., Amsterdam, 1994 | MR | Zbl

[4] M. Kesseböhmer, B. O. Stratmann, “Stern–Brocot pressure and multifractal spectra in ergodic theory of numbers”, Stoch. Dyn., 4:1 (2004), 77–84 | DOI | MR | Zbl

[5] N. Moshchevitin, A. Zhigljavsky, “Entropies of the partitions of the unit interval generated by the Farey tree”, Acta Arith., 115:1 (2004), 47–58 | DOI | MR | Zbl