Sums of nearest integer continued fractions with bounded digits: $\textrm{NICF}_5 + \textrm{NICF}_5 = {\mathbb R}$

Abstract

Adapting Cantor set methods that were used by Hall and Hlawka for regular continued fractions, we prove that every real number can be obtained as the sum of two real numbers for which the partial fractions in their nearest integer continued fraction expansion do not exceed 5 (with the zeroth partial fraction as the only possible exception). Furthermore, we prove that it is not possible to replace 5 by 4 in this result.