Cusick's conjecture on sums of continued-fraction Cantor sets

Let $S(k)$ be the set of numbers in $[0,1]$ whose continued-fraction partial quotients satisfy $a_n(x)\geq k$ whenever they are defined, with $0$ included. Cusick's conjecture. For $k\geq3$, the sumset $S(k)+S(k)$ has Lebesgue measure zero:

$$\lambda\bigl(S(k)+S(k)\bigr)=0.$$

The conjecture asserts that Cusick's theorem $S(2)+S(2)=[0,1]$ is unique in this measure-theoretic sense. The paper disproves it by proving that $S(k)+S(k)$ contains the interval $[0,1/(k-1)]$, and hence has positive Lebesgue measure for every $k\geq3$.