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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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