Every real number is a sum of two real numbers with diverging partial quotients

TL;DR

This paper proves that any irrational number can be expressed as the sum of two real numbers whose continued fraction partial quotients diverge, using a constructive proof based on a recently developed algorithm.

Contribution

It introduces a constructive method to decompose irrational numbers into sums of two numbers with diverging partial quotients, leveraging a novel algorithm.

Findings

Every irrational number can be represented as such a sum.

The proof utilizes a new algorithm by Nikita Shulga.

The study of the algorithm is of independent interest.

Abstract

We show that every irrational number is a sum of two real numbers with diverging partial quotients. The proof is constructive. The key towards these results is an algorithm which was recently developed by Nikita Shulga, and our study of this algorithm is of independent interest.