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.
"Martingale differences and the metric theory of continued fractions." Illinois J. Math. 52 (1) 213 - 242, Spring 2008. https://doi.org/10.1215/ijm/1242414129