In their work on homoclinic tangencies in dynamical systems, Palis and Takens raised several questions about arithmetic sums of pairs of dynamically defined and affine Cantor sets in the real line.
Let $C_1$ and $C_2$ be two affine Cantor sets in the real line and let $d_1$ and $d_2$ be their Hausdorff dimensions. It was shown by Peres and Solomyak that if $d_1+d_2 >1$, then "typically" the arithmetic sum $C_1 + C_2$ has positive Lebesgue measure. They also proved that if the sum of the dimensions is less than one, then typically the dimension of the sum equals $d_1+d_2$. It is natural to ask whether the corresponding Hausdorff measure of $C_1 + C_2$ is positive. We give a Bandt-Graf type necessary and sufficient condition for having zero $(d_1+d_2)$-dimensional Hausdorff measure of $C_1 + C_2$.
As a corollary, we obtain the result that if $C_1$ and $C_2$ are homogeneous affine Cantor sets of the same diameter, then the $(d_1+d_2)$-dimensional Hausdorff measure of $C_1 + C_2$ is always zero.