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On expressible sets of products. (English) Zbl 1340.11067
Let $$\{\alpha_n\}_{n\geq 1}$$ be a sequence of positive real numbers. Then the set $E_{\Pi}\{\alpha_n\}_{n\geq 1}=\left\{\prod_{n=1}^\infty\left(1+\frac{1}{\alpha_n c_n}\right)\mid c_n\in\mathbb{Z}^+\right\}$ is called its $$\Pi$$-expressible set, while the sequence $$\{\alpha_n\}_{n\geq 1}$$ is called $$\Pi$$-irrational if the elements of the set $$E_{\Pi}\{\alpha_n\}_{n\geq 1}$$ are irrational numbers.
The aim of the present paper is to calculate $$E_{\Pi}\{\alpha_n\}_{n\geq 1}$$ under various hypotheses on the sequence $$\{\alpha_n\}_{n\geq 1}$$. To this direction the authors managed to investigate specific results for the $$\Pi$$-expressible set of products as a continuation of related investigations on $$\Sigma$$-expressible sets of sums $E_{\Sigma}\{\alpha_n\}_{n\geq 1}=\left\{\sum_{n=1}^\infty \frac{1}{\alpha_n c_n}\mid c_n\in\mathbb{Z}^+\right\}.$
##### MSC:
 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11J72 Irrationality; linear independence over a field 11B05 Density, gaps, topology
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##### References:
 [1] Apéry, R, Irrationalité de $$ζ (2)$$ et $$ζ (3)$$, Asterisque, 61, 11-13, (1979) · Zbl 0401.10049 [2] Corvaja, P; Hančl, J, A transcendence criterion for infinite products, Atti Acad. Naz. Lincei. Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18, 295-303, (2007) · Zbl 1207.11075 [3] Erdős, P, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci., 10, 1-7, (1975) [4] Hančl, J, Expression of real numbers with the help of infinite series, Acta Arith. LIX., 2, 97-104, (1991) · Zbl 0701.11005 [5] Hančl, J; Kolouch, O, Erdős’ method for determining the irrationality of products, Bull. Aust. Math. Soc., 84, 414-424, (2011) · Zbl 1242.11050 [6] Hančl, J; Nair, R; Šustek, J, On the Lebesgue measure of the expressible sets of certain sequences, Indag. Math. N.S., 17, 567-581, (2006) · Zbl 1131.11048 [7] Kuhapatanakul, K; Laohakosol, V, Irrationality criteria for infinite products, J. Comb. Number Theory, 1, 49-57, (2008) · Zbl 1234.11093 [8] Luca, F; Tachiya, Y, Algebraic independence of infinite products generated by Fibonacci and Lucas numbers, Hokkaido Math. J., 43, 1-20, (2014) · Zbl 1291.11103 [9] K. Nishioka, Mahler functions and transcendence, Lecture Notes in Mathematics 1631, Springer, New York (1996) · Zbl 0876.11034 [10] Zhou, P; Lubinsky, DS, On the irrationality of $$∏ ^∞ _{j=0}(1± q^{-j}r+q^{-2j}s)$$, Analysis, 17, 129-153, (1997) · Zbl 0889.11025
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