Exponential diophantine equations and the irrationality of certain real numbers
Mahler proved the irrationality of the decimal fraction \(0.(1)(g)(g^ 2)..\). where \(g\geq 2\) is a fixed integer and \((g^ n)\) denotes the number \(g^ n\) in decimal form. Later, Shan and Wang proved that if g,h\(\geq 2\) are fixed integers and \((n_ k)_{k\in {\mathbb{N}}}\) a strictly increasing sequence of non-negative integers, then the positive real number (written in base h) \(0.(g^{n_ 1})_ h(g^{n_ 2})_ h...\) is irrational if we denote by \((a)_ h\) the number a written in base h. The author proves some further generalisations of which we give a typical example. Let \(a_ 1,...,a_ n\), \(u_ 1,...,u_ n\) be non-zero algebraic numbers such that \(u_ i/u_ j\) is not a root of unity for \(1\leq i<j\leq n\) and \((n_ k)_{k\in {\mathbb{N}}}\) a strictly increasing sequence of non- negative integers such that the numbers \(f_ k=\sum^{n}_{i=1}a_ iu_ i^{n_ k}\) are positive integers (e.g., the Fibonacci sequence). Let \(\theta =0\cdot (f_ 1)_ h(f_ 2)_ h(f_ 3)_ h..\) and denote by T(x) the number of digits of \((f_ 1)_ h(f_ 2)_ h...(f_ x)_ h.\) Then \[ | \theta -p/q| \quad >\quad h^{-T(C_ 1q^ 2)}\text{ for all } (p,q)\in {\mathbb{Z}}\times {\mathbb{N}}, \] where \(C_ 1\) is an effectively computable number depending only on the \(u_ i\) and h. The author also proves some results on exponential diophantine equations, the following of which is used in the proof of the above irrationality result: let \(a_ 1,...,a_ n\), \(u_ 1,...,u_ n\) (n\(\geq 2)\) be non- zero algebraic numbers such that \(u_ 1,...,u_ n\) and the numbers \(u_ i/u_ j\) \((1\leq i<j\leq n-1)\) are no roots of unity; then there exists an effectively computable number C, depending only on \(n,u_ 1,...,u_ n\), such that the equation \[ \sum^{n-1}_{i=1}a_ iu_ i^{x_ 1}+a_ nu_ n^{x_ 2}=1 \] has at most C solutions in \(x_ 1,x_ 2\in {\mathbb{Z}}\). The author derives this result from \textit{H. Schlickewei}'s explicit upper bound for the number of solutions of S-unit equations in several variables [Invent. Math. 102, No.1, 95-107 (1990; Zbl 0711.11017)].
scientific article; zbMATH DE number 4212138