On a Conjecture of Cusick on a sum of Cantor sets

In 1971 Cusick proved that every real number $x\in [0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$S(k):=\{x\in [0,1]:a_n(x)\ge k\text{ for all }n\in \mathbb{N}\},$$ then $$S(2)+S(2)=[0,1]\mathrm{.}$$ He also conjectured that this result is unique in the sense that if you exclude partial quotients from $1$ to $k-1$ with $k\ge 3$, then the Lebesgue measure $\lambda$ of the set of numbers which can be expressed as a sum of two continued fractions with no partial quotients from $\{1,\dots ,k-1\}$ is equal to $0$, that is $$\lambda (S(k)+S(k))=0\text{ for }k\ge 3.$$ In this paper, we disprove the conjecture of Cusick by showing that $$S(k)+S(k)\supseteq [0,\frac{1}{k-1}]\mathrm{.}$$ The proof is constructive and does not rely on ideas from previous works on the topic. We also show the existence of countably many ‘gaps’ in $S(k)+S(k)$, that is intervals, for which the endpoints lie in $S(k)+S(k)$, while none of the elements in the interior do so. Finally, we prove several results on the sums $$S(m)+S(n)$$ for $m\mathrm{\ne }n$.