Illinois Journal of Mathematics

Martingale differences and the metric theory of continued fractions

Alan K. Haynes and Jeffrey D. Vaaler

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We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special case of a class of martingale differences considered by Gundy. By applying known results for martingales, we obtain corresponding metric theorems for the continued fraction expansion of almost all real numbers.

Article information

Illinois J. Math., Volume 52, Number 1 (2008), 213-242.

First available in Project Euclid: 15 May 2009

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Zentralblatt MATH identifier

Primary: 11B57: Farey sequences; the sequences ${1^k, 2^k, \cdots}$ 11K50: Metric theory of continued fractions [See also 11A55, 11J70] 60G46: Martingales and classical analysis


Haynes, Alan K.; Vaaler, Jeffrey D. Martingale differences and the metric theory of continued fractions. Illinois J. Math. 52 (2008), no. 1, 213--242. doi:10.1215/ijm/1242414129.

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