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On expressible sets of geometric sequences. (English) Zbl 1215.11077
The paper under review is concerned with the set of real numbers $$x$$ which can be expressed in the form $x = \sum_{n=1}^\infty {1 \over {c_n A^n}},$ where $$A >3$$ is a real number and $$\{c_n\}$$ is some sequence of natural numbers. The set of such numbers is called the expressible set of the geometric sequence $$A^n$$.
It is shown that this set is Borel and contains the interval $$(0,1/((A-1)(\lceil A \rceil -2))]$$, and upper and lower bounds on the Lebesgue measure of the set are obtained. In the case when $$A = 4$$, the upper and lower bounds coincide, and the measure of the expressible set is equal to $$1/4$$. In this case, the interval shown to be contained in the set is equal to $$(0, 1/6]$$. Finally, the order at which the Lebesgue measure of the expressible set tends to zero as $$A$$ increases is studied. It is shown that a lower bound decays like $$A^{-2}$$, while an upper bound decays like $$A^{-3/2}$$. The exact asymptotic decay remains an open problem.

##### MSC:
 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
##### Keywords:
Expressible sets; geometric sequences; Lebesgue measure.
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