  This is a sequel to the paper arXiv:1312.6438 by the same authors. In this
sequel, we quantitatively improve several of the main results of
arXiv:1312.6438, and build on the methods therein.
  The main new results is that, for any finite set $A \subset \mathbb R$, there
exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$.
We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also,
we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg
\frac{|A|^2}{\log |A|}$. The latter result is optimal up to the logarithmic
factor.
