Results on sums of continued fractions

Abstract:

Let $F(m)$ be the (Cantor) set of infinite continued fractions with partial quotients no greater than m and let $F(m) + F(n) = \{ \alpha + \beta :\alpha \in F(m),\beta \in F(n)\}$. We show that $F(3) + F(4)$ is an interval of length 1.14 ... so every real number is the sum of an integer, an element of $F(3)$ and an element of $F(4)$. Similar results are given for $F(2) + F(7),F(2) + F(2) + F(4),F(2) + F(3) + F(3)$ and $F(2) + F(2) + F(2) + F(2)$. The techniques used are applicable to any Cantor sets in R for which certain parameters can be evaluated.