Linearly unrelated sequences. (English) Zbl 1005.11033

Hančl, Jaroslav

Pac. J. Math. 190, No. 2, 299-310 (1999).

The author generalizes a result of P. Erdős [J. Math. Sci. 10, 1-7 (1975; Zbl 0372.10023)] on irrationality of sums of infinite series as follows: Let \(\{a_{i,n}\}_{n=1}^\infty\) \((i=1,\dots, K)\) be sequences of positive real numbers. If for every sequence \(\{c_n\}_{n=1}^\infty\) of positive integers the numbers \(\sum_{n=1}^\infty 1/(a_{1,n}c_n),\dots, \sum_{n=1}^\infty 1/(a_{K,n}c_n)\), and 1 are linearly independent, then the sequences \(\{a_{i,n}\}_{n=1}^\infty\) \((i=1,\dots, K)\) are linearly unrelated. Then the author proves:
Theorem. Let \(\{a_{i,n}\}_{n=1}^\infty\), \(\{b_{i,n}\}_{n=1}^\infty\) \((i=1,\dots, K-1)\) be sequences of positive integers and \(\varepsilon> 0\) such that \[ \frac{a_{1,n+1}} {a_{1,n}}\geq 2^{K^{n-1}}, a_{1,n}|a_{1,n+1} \qquad (a_{1,n} \text{ divides }a_{1,n+1}), \tag{1} \]
\[ b_{i,n}< 2^{K^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}, \qquad i=1,\dots, K-1, \tag{2} \]
\[ \lim_{n\to\infty} \frac{a_{i,n} b_{j,n}} {b_{i,n} a_{j,n}}=0, \qquad \text{for all} \quad j,i\in \{1,\dots, K-1\},\;i>j, \tag{3} \]
\[ a_{i,n} 2^{-K^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}< a_{1,n}< a_{i,n} 2^{K^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}, \qquad i=1,\dots, K-1, \tag{4} \] hold for every sufficiently large natural number \(n\). Then the sequences \(\{a_{i,n}/ b_{i,n}\}_{n=1}^\infty\) \((i=1,\dots, K-1)\) are linearly unrelated.
Further, let \(\{A_n\}_{n=1}^\infty\) be a sequence of positive real numbers. If for every sequence \(\{c_n\}_{n=1}^\infty\) of positive integers the series \(\sum_{n=1}^\infty \frac{1}{A_nc_n}\) is irrational, then the sequence \(\{A_n\}_{n=1}^\infty\) is irrational. If \(\{A_n\}_{n=1}^\infty\) is not an irrational sequence, then it is a rational sequence. He proves
Theorem. Let \(\varepsilon>0\), and let \(\{a_n\}_{n=1}^\infty\) and \(\{b_n\}_{n=1}^\infty\) be two sequences of positive integers such that \(a_n\geq 2^{2^n}\) and \(b_n\leq 2^{2^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}\). Then the sequence \(\{\frac{\prod_{i=1}^n a_i}{b_n}\}_{n=1}^\infty\) is irrational and the series \(\sum_{n=1}^\infty \frac{b_n} {\prod_{n=1}^n a_i}\) is irrational too.
This theorem is an immediate consequence of the previous theorem. It is enough to put \(K=2\). From the last theorem he also obtains a criterion for Cantor sequences to be irrational.

Reviewer: O.Ninnemann (Berlin)

MSC:

11J72

Irrationality; linear independence over a field

Keywords:

linearly unrelated sequences; irrational sequences

Citations:

Zbl 0372.10023