| Title:
|
Diophantine Approximations of Infinite Series and Products (English) |
| Author:
|
Kolouch, Ondřej |
| Author:
|
Novotný, Lukáš |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
24 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
71-82 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos' results on~irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets. (English) |
| Keyword:
|
Infinite products |
| Keyword:
|
irrationality |
| Keyword:
|
linear independence |
| Keyword:
|
expressible set |
| MSC:
|
11J72 |
| MSC:
|
11J81 |
| MSC:
|
11K55 |
| idZBL:
|
Zbl 06670232 |
| idMR:
|
MR3546807 |
| . |
| Date available:
|
2016-08-26T11:22:56Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145806 |
| . |
| Reference:
|
[1] Badea, C.: The irrationality of certain infinite products.Studia Univ. Babeş-Bolyai Math., 31, 3, 1986, 3-8, Zbl 0625.10027, MR 0911859 |
| Reference:
|
[2] Badea, C.: The irrationality of certain infinite series.Glasgow Mathematical Journal, 29, 2, 1987, 221-228, Zbl 0629.10027, MR 0901668, 10.1017/S0017089500006868 |
| Reference:
|
[3] Duverney, D.: Sur les séries de nombres rationnels à convergence rapide.Comptes Rendus de l'Académie des Sciences, Series I, Mathematics, 328, 7, 1999, 553-556, Zbl 0940.11027, MR 1680014 |
| Reference:
|
[4] Erdős, P.: Problem 4321.The American Mathematical Monthly, 64, 7, 1950, |
| Reference:
|
[5] Erdős, P.: Some Problems and Results on the Irrationality of the Sum of Infinite Series.Journal of Mathematical Sciences, 10, 1975, 1-7, Zbl 0372.10023, MR 0539489 |
| Reference:
|
[6] Erdős, P.: Erdős problem no. 6.1995 Prague Midsummer Combinatorial Workshop, KAM Series (95-309) (ed. M. Klazar) (KAM MPP UK, Prague, 1995), 1995, |
| Reference:
|
[7] Erdős, P., Straus, E. G.: On the irrationality of certain Ahmes series.Journal of Indian Mathematical Society, 27, 1964, 129-133, MR 0175848 |
| Reference:
|
[8] Hančl, J.: Expression of Real Numbers with the Help of Infinite Series.Acta Arithmetica, LIX, 2, 1991, 97-104, Zbl 0701.11005, MR 1133951, 10.4064/aa-59-2-97-104 |
| Reference:
|
[9] Hančl, J.: Criterion for Irrational Sequences.Journal of Number Theory, 43, 1, 1993, 88-92, Zbl 0768.11021, MR 1200812, 10.1006/jnth.1993.1010 |
| Reference:
|
[10] Hančl, J., Filip, F.: Irrationality Measure of Sequences.Hiroshima Math. J., 35, 2, 2005, 183-195, Zbl 1087.11049, MR 2176050 |
| Reference:
|
[11] Hančl, J., Kolouch, O.: Erdős' method for determining the irrationality of products.Bull. Aust. Math. Soc., 84, 3, 2011, 414-424, Zbl 1242.11050, MR 2851961, 10.1017/S0004972711002309 |
| Reference:
|
[12] Hančl, J., Kolouch, O.: Irrationality of infinite products.Publ. Math. Debrecen, 83, 4, 2013, 667-681, Zbl 1299.11050, MR 3150834, 10.5486/PMD.2013.5676 |
| Reference:
|
[13] Hančl, J., Kolouch, O., Novotný, L.: A Criterion for linear independence of infinite products.An. St. Univ. Ovidius Constanta, 23, 2, 2015, 107-120, Zbl 1349.11105, MR 3348703 |
| Reference:
|
[14] Hančl, J., Kolouch, O., Pulcerová, S., Štěpnička, J.: A note on the transcendence of infinite products.Czechoslovak Math. J., 62, 137, 2012, 613-623, Zbl 1265.11078, MR 2984622, 10.1007/s10587-012-0053-2 |
| Reference:
|
[15] Hančl, J., Korčeková, K., Novotný, L.: Productly linearly independent sequences.Stud. Sci. Math. Hung., 52, 2015, 350-370, Zbl 1363.11073, MR 3402910 |
| Reference:
|
[16] Hančl, J., Nair, R., Novotný, L.: On expressible sets of products.Period. Math. Hung., 69, 2, 2014, 199-206, Zbl 1340.11067, MR 3278957, 10.1007/s10998-014-0058-8 |
| Reference:
|
[17] Hančl, J., Nair, R., Novotný, L., Šustek, J.: On the Hausdorff dimension of the expressible set of certain sequences.Acta Arithmetica, 155, 1, 2012, 85-90, Zbl 1272.11094, MR 2982430, 10.4064/aa155-1-8 |
| Reference:
|
[18] Hančl, J., Nair, R., Šustek, J.: On the Lebesgue measure of the expressible set of certain sequences.Indag. Mathem., 17, 4, 2006, 567-581, Zbl 1131.11048, MR 2320114, 10.1016/S0019-3577(06)81034-7 |
| Reference:
|
[19] Hančl, J., Schinzel, A., Šustek, J.: On Expressible Sets of Geometric Sequences.Funct. Approx. Comment. Math., 38, 2008, 341-357, Zbl 1215.11077, MR 2490089 |
| Reference:
|
[20] Kurosawa, T., Tachiya, Y., Tanaka, T.: Algebraic independence of the values of certain infinite products and their derivatives related to Fibonacci and Lucas numbers (Analytic Number Theory: Number Theory through Approximation and Asymptotics).Proceedings of Institute for Mathematical Sciences, Kyoto University, 1874, 2014, 81-93, Research Institute for Mathematical Sciences, MR 3178476 |
| Reference:
|
[21] Luca, F., Tachiya, Y.: Algebraic independence of infinite products generated by Fibonacci and Lucas numbers.Hokkaido Math. J., 43, 2014, 1-20, Zbl 1291.11103, MR 3178476, 10.14492/hokmj/1392906090 |
| Reference:
|
[22] Nyblom, M. A.: On the construction of a family of transcendental valued infinite products.Fibonacci Quart., 42, 4, 2004, 353-358, Zbl 1062.11048, MR 2110089 |
| Reference:
|
[23] Sándor, J.: Some classes of irrational numbers.Studia Universitatis Babeş-Bolyai Mathematica, 29, 1984, 3-12, Zbl 0544.10033, MR 0782282 |
| Reference:
|
[24] Väänänen, K.: On the approximation of certain infinite products.Math. Scand., 73, 2, 1993, 197-208, Zbl 0818.11028, MR 1269258, 10.7146/math.scand.a-12465 |
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