  This paper is a complement to our previous paper [21]. It surveys the works
on the Furstenberg set $S=\{2^{m}3^{n}: n\ge 0, m\ge 0\}$ and its random
version $T$. We also present some new results. For example, it is proved that
$T$ almost surely contains a subset of positive lower density which is
$\frac{4}{3}$-Rider. It is also proved that a class of random sets of integers
are Sidon sets when Bourgain's condition is not satisfied; this generalizes a
result of Kahane-Katznelson. Some open questions about $S$ and $T$ are listed
at the end of the paper.
