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 \(\mu (M(g))\leq 2g^ 2/(g-1)\).


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