Cantor sets and numbers with restricted partial quotients

Abstract:

For $j=1,\dots ,k$ let $C_j$ be a Cantor set constructed from the interval $I_j$, and let $\epsilon _j=\pm 1$. We derive conditions under which \begin{equation*} \epsilon _1 C_1+\dots +\epsilon _k C_k = \epsilon _1 I_1+\dots +\epsilon _k I_k \quad \text {and}\quad C_1^{\epsilon _1}\dotsb C_k^{\epsilon _k}= I_1^{\epsilon _1}\dotsb I_k^{\epsilon _k}.\end{equation*} When these conditions do not hold, we derive a lower bound for the Hausdorff dimension of the above sum and product. We use these results to make corresponding statements about the sum and product of sets $F(B_j)$, where $B_j$ is a set of positive integers and $F(B_j)$ is the set of real numbers $x$ such that all partial quotients of $x$, except possibly the first, are members of $B_j$.