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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
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