Two concrete new constructions of the real numbers
For the Engel and Sylvester expansions [see the reviewer, Representations of real numbers by infinite series (Lect. Notes Math. 502) (Berlin: Springer 1976; Zbl 0322.10002)] the authors establish a criterion for monotonicity by means of the digits, from which they deduce that the formal series for rationals are dense in the set of all expansions. From these facts, irrationals are introduced as limits of monotonic sequences of representations signifying rationals, thus presenting the real numbers via the mentioned two expansions. [Reviewer's remark: Since the monotonicity criterion established here for the Engel and Sylvester expansions has been known for a much larger class of expansions (see Lemma 1.4 -- all references are to the above cited book), upon covering the real line by disjoint \(N\)-th order cylinders (see pp. 6--7), one immediately gets that the expansions signifying rationals are dense. Furthermore, for the Oppenheim series for which \(h_ j(d_ j)\) is an integer multiple of \(d_ j-1\), rationality is characterized by \(d_{j+1}=h_ j(d_ j)+1\) for all large j (see pp. 14--15, 25 and 28), and thus all ``real numbers'' can be approximated by ``rationals with \(N\) digits'' at a faster rate than any exponential function (see Theorem 1.8 and use the fact that \(d_ N(x)=k_ N\geq D_ N\), where the constant sequence \(D_ n\) is defined by \(D_ 1=2\), \(D_{n+1}=h_ n(D_ n)+1\), \(n\geq 1)\). Note that the assumption on \(h_ j(n)\) is satisfied by Engel and Sylvester series and by Cantor's product (see pp. 17--18).]
scientific article; zbMATH DE number 4108850