On expressible sets of geometric sequences. (English) Zbl 1215.11077

Hančl, Jaroslav; Schinzel, Andrzej; Šustek, Jan

Funct. Approximatio, Comment. Math. 39, Part 1, 71-95 (2008).

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.

Reviewer: Simon Kristensen (Aarhus)

MSC:

11K55

Metric theory of other algorithms and expansions; measure and Hausdorff dimension

Keywords:

Expressible sets; geometric sequences; Lebesgue measure.