Cantor sets and numbers with restricted partial quotients

Abstract

For j = 1, ..., k let Cj be a Cantor set constructed from the interval Ij, and let Ej = =/1. We derive conditions under which E1C1 + ... + EkCk = E1I1 +...+ EkIk and C1e1 ... Cekk = Ie1i ... Iekh. When these conditions do not hold, we derive a lower bound for the Hausdorff dimensions of the above sum and product. We use these results to make corresponding statements about the sum and product of sets F(Bj), where Bj is a set of positive integers and F(Bj) is the set of real numbers x such that all partial quotients of x, except possibly the first, are members of Bj. We also examine cases where our conditions do not hold, but in which it is still the case that C1+C2 contains an interval.