Martingale differences and the metric theory of continued fractions

TL;DR

This paper explores the connection between martingale differences and continued fraction expansions, deriving metric theorems for almost all real numbers through martingale theory.

Contribution

It introduces a novel orthonormal system encoding continued fractions as martingale differences, linking martingale theory with metric properties of continued fractions.

Findings

Established a complete system of martingale differences related to continued fractions

Derived metric theorems for the distribution of continued fraction expansions

Connected martingale results to properties of almost all real numbers

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 R. F. Gundy. By applying known results for martingales we obtain corresponding metric theorems for the continued fraction expansion of almost all real numbers.