New normality constructions for continued fraction expansions

TL;DR

This paper introduces new methods for constructing continued fraction normal numbers, including sequences based on primes, expanding the class of known normality constructions in continued fractions.

Contribution

It presents novel constructions of continued fraction normal numbers, notably using subsequences of rationals with prime numerators and denominators.

Findings

New continued fraction normal numbers constructed from prime-based sequences.

Extension of normality results to broader classes of rational sequences.

Demonstration of normality for specific subsequences of rationals.

Abstract

Adler, Keane, and Smorodinsky showed that if one concatenates the finite continued fraction expansions of the sequence of rationals \[ \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \cdots \] into an infinite continued fraction expansion, then this new number is normal with respect to the continued fraction expansion. We show a variety of new constructions of continued fraction normal numbers, including one generated by the subsequence of rationals with prime numerators and denominators: \[ \frac{2}{3}, \frac{2}{5}, \frac{3}{5}, \frac{2}{7}, \frac{3}{7}, \frac{5}{7},\cdots. \]