Illinois Journal of Mathematics

Martingale differences and the metric theory of continued fractions

Alan K. Haynes and Jeffrey D. Vaaler

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Abstract

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

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

Dates
First available in Project Euclid: 15 May 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1242414129

Digital Object Identifier
doi:10.1215/ijm/1242414129

Mathematical Reviews number (MathSciNet)
MR2507242

Zentralblatt MATH identifier
1236.11068

Subjects
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

Citation

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. https://projecteuclid.org/euclid.ijm/1242414129


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