Geodesic
On the derivative of the Minkowski question mark function~$?(x)$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 33-44.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $x=[0;a_1,a_2,\dots]$ be the regular continued fraction expansion an irrational number $x\in[0,1]$. For the derivative of the Minkowski function $?(x)$ we prove that $?'(x)=+\infty$, provided that $\limsup_{t\to\infty}\frac{a_1+\dots+a_t}t\kappa_1=\frac{2\log\lambda_1}{\log 2} = 1.388^+$, and $?'(x) = 0$, provided that $\liminf\limits_{t\to \infty}\frac{a_1+\dots+a_t}t>\kappa_2=\frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$, where $L_j=\log\bigl(\frac{j+\sqrt{j^2+4}}2\bigr)-j\cdot\frac{\log2}2$. Constants $\kappa_1$, $\kappa_2$ are the best possible. It is also shown that $?'(x)=+\infty$ for all $x$ with partial quotients bounded by $4$. 
@article{FPM_2010_16_6_a3,
     author = {A. A. Dushistova and N. G. Moshchevitin},
     title = {On the derivative of the {Minkowski} question mark function~$?(x)$},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {33--44},
     publisher = {mathdoc},
     volume = {16},
     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a3/}
}
TY  - JOUR
AU  - A. A. Dushistova
AU  - N. G. Moshchevitin
TI  - On the derivative of the Minkowski question mark function~$?(x)$
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2010
SP  - 33
EP  - 44
VL  - 16
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a3/
LA  - ru
ID  - FPM_2010_16_6_a3
ER  - 
%0 Journal Article
%A A. A. Dushistova
%A N. G. Moshchevitin
%T On the derivative of the Minkowski question mark function~$?(x)$
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2010
%P 33-44
%V 16
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a3/
%G ru
%F FPM_2010_16_6_a3
A. A. Dushistova; N. G. Moshchevitin. On the derivative of the Minkowski question mark function~$?(x)$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 33-44. http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a3/

[1] Jenkinson O., “On the density of Hausdorff dimension of bounded type continued fraction sets: the Texan conjecture”, Stoch. Dyn., 4 (2004), 63–76 | DOI | MR | Zbl

[2] Kan I. D., “Refining of the comparison rule for continuants”, Discrete Math. Appl., 10:5 (2000), 477–480 | DOI | MR | Zbl

[3] Kesseböhmer M., Stratmann B. O., Fractal analysis for sets of nondifferentiability of Minkowski question mark function, 2007, arXiv: 0706.0453v1[math.DS]

[4] Kinney J. R., “Note on a singular function of Minkowski”, Proc. Am. Math. Soc., 11 (1960), 788–789 | DOI | MR

[5] Minkowski H., Gesammelte Abhandlungen, v. 2, 1911

[6] Motzkin T. S., Straus E. G., “Some combinatorial extremum problems”, Proc. Am. Math. Soc., 7 (1956), 1014–1021 | DOI | MR

[7] Paradis J., Viader P., Bibiloni L., “A new light on Minkowski's $?(x)$ function”, J. Number Theory, 73 (1998), 212–227 | DOI | MR | Zbl

[8] Paradis J., Viader P., Bibiloni L., “The derivative of Minkowski's $?(x)$ function”, J. Math. Anal. Appl., 253 (2001), 107–125 | DOI | MR | Zbl

[9] Salem R., “On some singular monotone functions which are strictly increasing”, Trans. Am. Math. Soc., 53 (1943), 427–439 | DOI | MR | Zbl