Sums of sets of continued fractions

Abstract:

For each integer $k \geqq 2$, let $S(k)$ denote the set of real numbers $\alpha$ such that $0 \leqq \alpha \leqq {k^{ - 1}}$ and $\alpha$ has a continued fraction containing no partial quotient less than k. It is proved that every number in the interval [0, 1] is representable as a sum of k elements of $S(k)$.