Explicit continued fractions with expected partial quotient growth (Q2781326)
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scientific article; zbMATH DE number 1721070
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Explicit continued fractions with expected partial quotient growth |
scientific article; zbMATH DE number 1721070 |
Statements
Explicit continued fractions with expected partial quotient growth (English)
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19 March 2002
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continued fractions
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measure theory
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For \(0< x< 1\), let \([0, \alpha_1(x), \alpha_2(x),\dots]\) be the continued fraction expansion of \(x\), and put \(L_N(x)= \max_{1\leq n\leq N} a_n(x)\). A result of \textit{W. Philipp} [Acta Arith. 28, 379--386 (1976; Zbl 0332.10033)] says that \(\liminf_{N\to \infty} \frac 1N\cdot L_N(x)\cdot \log\log N= \frac{1}{\log 2}\) holds for almost all \(x\). The author gives a sufficient criterion for and examples of numbers \(x\) which satisfy this relation.
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