  Let $B$ be a set of natural numbers of size $n$. We prove that the length of
the longest arithmetic progression contained in the product set $B.B = \{bb'|
\, b, b' \in B\}$ cannot be greater than $O(n \log n)$ which matches the lower
bound provided in an earlier paper up to a multiplicative constant. For sets of
complex numbers we improve the bound to $O_\epsilon(n^{1 + \epsilon})$ for
arbitrary $\epsilon > 0$ assuming the GRH.
