On the derivative of the Minkowski question mark function $?(x)$
| dc.creator | Dushistova, Anna A. | |
| dc.creator | Moshchevitin, Nikolai G. | |
| dc.date | 2007-06-15 | |
| dc.date | 2007-12-17 | |
| dc.date.accessioned | 2026-07-07T08:49:10Z | |
| dc.date.available | 2026-07-07T08:49:10Z | |
| dc.description | Let $ x = [0;a_1,a_2,...]$ be the decomposition of the irrational number $x \in [0,1]$ into regular continued fraction. Then for the derivative of the Minkowski function $?(x)$ we prove that $?'(x) = +\infty$ provided $ \limsup_{t\to \infty}\frac{a_1+...+a_t}{t} <κ_1 =\frac{2\log λ_1}{\log 2} = 1.388^+$, and $?'(x) = 0$ provided $ \liminf_{t\to \infty}\frac{a_1+...+a_t}{t} >κ_2 = \frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$ (here $ L_j = \log (\frac{j+\sqrt{j^2+4}}{2}) - j\cdot\frac{\log 2}{2}$). Constants $κ_1,κ_2$ are the best possible. Also we prove that $?'(x) = +\infty$ holds for all $x$ with partial quotients bounded by 4. | |
| dc.description | 10 pages, submitted to Discrete Mathematics and Applications, minor correction of misprints | |
| dc.identifier | https://arxiv.org/abs/0706.2219 | |
| dc.identifier | http://arxiv.org/abs/0706.2219 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/144202 | |
| dc.subject | Number Theory | |
| dc.subject | 11J70, 11J83 | |
| dc.title | On the derivative of the Minkowski question mark function $?(x)$ | |
| dc.type | text |