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Metric theory of Pierce expansions. (English) Zbl 0598.10057
Every real number $$0<x\leq 1$$ has a unique representation of the form $$x=1/a_ 1-1/a_ 1a_ 2+1/a_ 1a_ 2a_ 3-...,$$ where the integers $$a_ j$$ are determined by the algorithm $$x=x_ 0$$, $$a_ j=[1/x_{j- 1}]$$ and $$x_ j=1-a_ jx_{j-1}$$, with $$[y]$$ signifying the integer part of $$y$$. This algorithm, and the related representation of $$x$$, is known as Pierce’s expansion. The author investigates the metric properties of the sequence $$a_ j$$, when the underlying measure is that of Lebesgue. Connection with Stirling numbers arises, and simple proofs are found for several neat identities, some of which are new. Several properties of the $$a_ j=a_ j(x)$$ are remarkably similar to the coefficients in Engel’s series [see, e.g., the reviewer’s monograph “Representations of real numbers by infinite series” (Springer Lect. Notes Math. 502) (1976; Zbl 0322.10002)].

##### MSC:
 11K50 Metric theory of continued fractions 11B73 Bell and Stirling numbers 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 05A19 Combinatorial identities, bijective combinatorics