Approximations of real numbers by the sequence \(\{\) n\(\alpha\) \(\}\) and their metrical theory
For an irrational real number \(0<\alpha <1\), let \([a_ 1,a_ 2,...]\) be its simple continued fraction expansion. Set \(p_ n/q_ n=[a_ 1,a_ 2,...,a_ n].\) Then any real number \(-\alpha <\beta <1-\alpha\) can uniquely be represented as \[ \beta =\sum^{+\infty}_{i=1}b_ i(\beta)(q_{i-1}\alpha -p_{i-1}), \] where, for \(b_ i=b_ i(\beta)\), the inequalities \(0\leq b_ 1\leq a_ 1-1\) and \(0\leq b_ i\leq a_ i\), \(i\geq 2\), hold (if \(b_ i=a_ i\) then \(b_{i-1}=0\) for \(i\geq 2).\) The authors prove a number of remarkable limit theorems. Two examples: (i) Set \(u_ n=\sum^{n}_{i=1}b_ i(\beta)q_{i-1}.\) Then, for almost all (\(\alpha\),\(\beta)\), \((\log u_ n)/n \to \pi^ 2/12 \log 2\) (which is known to be the (almost everywhere) limit of (log \(q_ n)/n)\). (ii) Let \(N\nu_ N\) be the number of integers \(n\leq N\) such that \[ | \beta -\sum^{n}_{i=1}b_ i(\beta)(q_{i-1}\alpha -p_{i- 1})| \leq z/q_ n. \] Then \(\nu_ N\to z(2-z)\), \(0\leq z\leq 1\).
scientific article; zbMATH DE number 4073091