Sub-self-similar sets with distinct Hausdorff and Box dimensions

We construct an IFS consisting of three contracting similarities on ${\mathbb{R}}^9$ with a countable compact sub-self-similar set $E$ with distinct Hausdorff and lower box dimensions. This provides a concrete counterexample to the question raised by Falconer concerning the equality of Hausdorff and box dimensions for sub-self-similar sets without the open set condition.