On a Conjecture of Cusick on a sum of Cantor sets

Abstract

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 $$ S(k):= \{ x\in[0,1] : a_n(x) \geq k \text{ for all } n\in\mathbb{N} \}, $$ then $$ S(2)+S(2) = [0,1]. $$ He also conjectured that this result is unique in the sense that if you exclude partial quotients from $1$ to $k-1$ with $k\geq3$, 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 $\{1,\ldots,k-1\}$ is equal to $0$, that is $$\lambda\Bigl( S(k)+S(k) \Bigl)= 0 \text{ for }k\geq 3.$$ In this paper, we disprove the conjecture of Cusick by showing that $$ S(k)+S(k) \supseteq \left[0,\frac{1}{k-1}\right]. $$ 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 $m\neq n$.