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Keywords:
Infinite products; irrationality; linear independence; expressible set
Infinite products; irrationality; linear independence; expressible set
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.
References:
[1] Badea, C.: The irrationality of certain infinite products. Studia Univ. Babeş-Bolyai Math., 31, 3, 1986, 3-8, MR 0911859 | Zbl 0625.10027
[2] Badea, C.: The irrationality of certain infinite series. Glasgow Mathematical Journal, 29, 2, 1987, 221-228, DOI 10.1017/S0017089500006868 | MR 0901668 | Zbl 0629.10027
[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, MR 1680014 | Zbl 0940.11027
[4] Erdős, P.: Problem 4321. The American Mathematical Monthly, 64, 7, 1950,
[5] Erdős, P.: Some Problems and Results on the Irrationality of the Sum of Infinite Series. Journal of Mathematical Sciences, 10, 1975, 1-7, MR 0539489 | Zbl 0372.10023
[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,
[7] Erdős, P., Straus, E. G.: On the irrationality of certain Ahmes series. Journal of Indian Mathematical Society, 27, 1964, 129-133, MR 0175848
[8] Hančl, J.: Expression of Real Numbers with the Help of Infinite Series. Acta Arithmetica, LIX, 2, 1991, 97-104, DOI 10.4064/aa-59-2-97-104 | MR 1133951 | Zbl 0701.11005
[9] Hančl, J.: Criterion for Irrational Sequences. Journal of Number Theory, 43, 1, 1993, 88-92, DOI 10.1006/jnth.1993.1010 | MR 1200812 | Zbl 0768.11021
[10] Hančl, J., Filip, F.: Irrationality Measure of Sequences. Hiroshima Math. J., 35, 2, 2005, 183-195, MR 2176050 | Zbl 1087.11049
[11] Hančl, J., Kolouch, O.: Erdős' method for determining the irrationality of products. Bull. Aust. Math. Soc., 84, 3, 2011, 414-424, DOI 10.1017/S0004972711002309 | MR 2851961 | Zbl 1242.11050
[12] Hančl, J., Kolouch, O.: Irrationality of infinite products. Publ. Math. Debrecen, 83, 4, 2013, 667-681, DOI 10.5486/PMD.2013.5676 | MR 3150834 | Zbl 1299.11050
[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, MR 3348703 | Zbl 1349.11105
[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, DOI 10.1007/s10587-012-0053-2 | MR 2984622 | Zbl 1265.11078
[15] Hančl, J., Korčeková, K., Novotný, L.: Productly linearly independent sequences. Stud. Sci. Math. Hung., 52, 2015, 350-370, MR 3402910 | Zbl 1363.11073
[16] Hančl, J., Nair, R., Novotný, L.: On expressible sets of products. Period. Math. Hung., 69, 2, 2014, 199-206, DOI 10.1007/s10998-014-0058-8 | MR 3278957 | Zbl 1340.11067
[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, DOI 10.4064/aa155-1-8 | MR 2982430 | Zbl 1272.11094
[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, DOI 10.1016/S0019-3577(06)81034-7 | MR 2320114 | Zbl 1131.11048
[19] Hančl, J., Schinzel, A., Šustek, J.: On Expressible Sets of Geometric Sequences. Funct. Approx. Comment. Math., 38, 2008, 341-357, MR 2490089 | Zbl 1215.11077
[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
[21] Luca, F., Tachiya, Y.: Algebraic independence of infinite products generated by Fibonacci and Lucas numbers. Hokkaido Math. J., 43, 2014, 1-20, DOI 10.14492/hokmj/1392906090 | MR 3178476 | Zbl 1291.11103
[22] Nyblom, M. A.: On the construction of a family of transcendental valued infinite products. Fibonacci Quart., 42, 4, 2004, 353-358, MR 2110089 | Zbl 1062.11048
[23] Sándor, J.: Some classes of irrational numbers. Studia Universitatis Babeş-Bolyai Mathematica, 29, 1984, 3-12, MR 0782282 | Zbl 0544.10033
[24] Väänänen, K.: On the approximation of certain infinite products. Math. Scand., 73, 2, 1993, 197-208, DOI 10.7146/math.scand.a-12465 | MR 1269258 | Zbl 0818.11028
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