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Approximation to certain transcendental decimal fractions by algebraic numbers. (English) Zbl 0718.11030

For a positive integer \(g\geq 2\) consider the number \(M(g)=0.(1)_ g(2)_ g...(n)_ g...\) where for \(n\in {\mathbb{N}}\) \((n)_ g\) means digit representation of n to base g and \(0.(1)_ g(2)_ g...\) means digit representation of the real M(g) to base g. It is known that M(g) is transcendental. The author gives estimates for Mahler’s function \(w_ d\) and for Koksma’s function \(w^*_ d\) of this number and he determines the exact value of the irrationality measure \(\mu (M(g)):=w_ 1(M(g))+1=w^*_ 1(M(g))+1\) of M(g), namely \(\mu M(g))=g\) for any \(g\geq 2\). Until now it was only known that \(\mu (M(g))\leq 2g^ 2/(g-1)\).

MSC:

11J82 Measures of irrationality and of transcendence
11J17 Approximation by numbers from a fixed field
11A63 Radix representation; digital problems
Full Text: DOI

References:

[1] Alladi, K.; Robinson, M. L., Legendre polynomials and irrationality, J. Reine angew. Math., 318, 135-155 (1980) · Zbl 0425.10039
[2] Danilov, L. V., Rational approximations of some functions at rational points, Math. Notes USSR, 24, 741-746 (1979) · Zbl 0418.10033
[3] Mahler, K., Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Nederl. Akad. Wetensch. Proc. Ser. A., 40, 421-428 (1937) · JFM 63.0156.01
[4] Mahler, K., On a class of transcendental decimal fractions, Comm. Pure Appl. Math., 29, 717-725 (1976) · Zbl 0339.10025
[5] Schneider, Th., Einführung in die transzendenten Zahlen (1957), Springer-Verlag: Springer-Verlag Berlin · Zbl 0077.04703
[6] Shallit, J. O., Simple continued fractions for some irrational numbers, II, J. Number Theory, 14, 228-231 (1982) · Zbl 0481.10005
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