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The Ostrogradsky series and related Cantor-like sets. (English) Zbl 1131.11052
The authors study the Ostrogradsky representation of real numbers, i.e., the expansion of a real number \(x\) in the following form:
\[ \begin{aligned} x &= \sum_n \frac{(-1)^{n-1}}{q_1 \ldots q_n}\\ &= \sum_n\frac{(-1)^{n-1}}{g_1(g_1+g_2)\ldots(g_1+\cdots+g_n)} \equiv \overline{\text{O}}(g_1,\ldots,g_n,\ldots),\end{aligned} \]
where \(q_{n+1}>q_n\in\mathbb N\), \(g_1=q_1\), \(g_{n+1}=q_{n+1}-q_n\). They compare this representation with the corresponding one in terms of continued fractions. They establish basic metric relations (equalities and inequalities for ratios of the length of cylindrical sets). They also study properties of subsets belonging to some classes of closed nowhere dense sets defined by characteristic properties of the \(\overline{O}\)-representation. In particular, conditions for the set \(C[\overline{O},V]\), consisting of real numbers whose \(\overline{O}\)-symbols take values from the set \(V \subset \mathbb N\), to be of zero resp. positive Lebesgue measure are found.
Please note that such expansions are sometimes called Pierce series in the literature (the authors mention a paper of Pierce, see also, e.g. J. Paradis, P. Viader and L. Bibiloni [Am. Math. Mon. 106, No. 3, 241–251 (1999; Zbl 0994.11013); Acta Arith. 91, No. 2, 107–115 (1999; Zbl 0982.11044); Fibonacci Q. 36, No. 2, 146–153 (1998; Zbl 0911.11007)], J. Shallit [Fibonacci Q. 32, No. 6, 416–423 (1994; Zbl 0823.11043); ibid. 24, 22–40 (1986; Zbl 0598.10057); ibid. 22 332–335 (1984; Zbl 0546.10005)], P. Erdős and J. O. Shallit [Sémin. Théor. Nombres Bordx., Sér. II 3, No. 1, 43–53 (1991; Zbl 0727.11003)]).

MSC:
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11K50 Metric theory of continued fractions
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