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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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