Variations on the sum-product problem II

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 $\mathrm{\mid }A(A+a)\mathrm{\mid }\gtrsim \mathrm{\mid }A{\mathrm{\mid }}^{\frac{3}{2}+\frac{1}{186}}$. 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.