| Title:
|
A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself (English) |
| Author:
|
Marques, Diego |
| Author:
|
Silva, Elaine |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
25 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
1-4 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note, we prove that there is no transcendental entire function $f(z)\in \mathbb{Q} [[z]]$ such that $f(\mathbb{Q} )\subseteq \mathbb{Q}$ and $\mathop{\rm den} f(p/q)=F(q)$, for all sufficiently large $q$, where $F(z)\in \mathbb{Z} [z]$. (English) |
| Keyword:
|
Liouville numbers |
| Keyword:
|
Mahler's question |
| Keyword:
|
power series |
| MSC:
|
11J81 |
| idZBL:
|
Zbl 06888083 |
| idMR:
|
MR3667071 |
| . |
| Date available:
|
2017-09-01T12:08:50Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146837 |
| . |
| Reference:
|
[1] Mahler, K.: Arithmetic properties of lacunary power series with integral coefficients.J. Austral. Math. Soc., 5, 1965, 56-64, Zbl 0148.27703, MR 0190094, 10.1017/S1446788700025866 |
| Reference:
|
[2] Mahler, K.: Some suggestions for further research.Bull. Austral. Math. Soc., 29, 1984, 101-108, Zbl 0517.10001, MR 0732177, 10.1017/S0004972700021316 |
| Reference:
|
[3] Maillet, E.: Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions.1906, Gauthier-Villars, Paris, |
| Reference:
|
[4] Marques, D., Moreira, C.G.: A variant of a question proposed by K. Mahler concerning Liouville numbers.Bull. Austral. Math. Soc., 91, 2015, 29-33, Zbl 1308.11069, MR 3294255, 10.1017/S0004972714000471 |
| Reference:
|
[5] Marques, D., Ramirez, J.: On transcendental analytic functions mapping an uncountable class of $U$-numbers into Liouville numbers.Proc. Japan Acad. Ser. A Math. Sci., 91, 2015, 25-28, Zbl 1311.11067, MR 3310967, 10.3792/pjaa.91.25 |
| Reference:
|
[6] Marques, D., Ramirez, J., Silva, E.: A note on lacunary power series with rational coefficients.Bull. Austral. Math. Soc., 93, 2015, 1-3, MR 3491477 |
| Reference:
|
[7] Marques, D., Schleischitz, J.: On a problem posed by Mahler.J. Austral. Math. Soc., 100, 2016, 86-107, Zbl 1335.11053, MR 3436773, 10.1017/S1446788715000415 |
| . |
