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On the Hausdorff dimension of the expressible set of certain sequences. (English) Zbl 1272.11094
The expressible set of a real sequence $$\{a_n\}$$ consists of all numbers $$x$$ of the form $x = \sum_{n=0}^\infty {1 \over {a_n c_n}}, \quad c_n \in {\mathbb N}.$ In the present paper, it is shown that if two real sequences $$\{a_n\}$$ and $$\{b_n\}$$ satisfy the conditions that for some $$\varepsilon, \delta > 0$$ and some $$\alpha \in (0,1)$$, $\limsup_{n \rightarrow \infty} a_n^{1/(3+\delta)^n} = \infty, \quad a_n \geq n^{1+\varepsilon}, \quad b_n \leq 2^{\log_2^\alpha a_n},$ for all $$n$$ sufficiently large, then the expressible set of the sequence $$\{a_n/b_n\}$$ has Hausdorff dimension at most $$2/(2+\delta)$$.
In fact, the authors prove a considerably more technical result, which in turn is proved by a covering argument together with a lemma due to J. Hančl, R. Nair and J. Šustek [Indag. Math., New Ser. 17, No. 4, 567–581 (2006; Zbl 1131.11048)]. The main theorem follows directly from this result.
##### MSC:
 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11J72 Irrationality; linear independence over a field
##### Keywords:
expressible set; Hausdorff dimension
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