  We prove new results on additive properties of finite sets $A$ with small
multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as
well as multiplicative subgroups in the prime residue field. The improvements
are based on new combinatorial lemmata, which may be of independent interest.
  Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$
over the reals, "redistributing" the exponents in the textbook Elekes
sum-product inequality and the new best known additive energy bound $\mathsf
E(A)\lesssim_M |A|^{49/20}$, which aligns, in a sense to be discussed, with the
best known sum set bound $|A+A|\gtrsim_M |A|^{8/5}$.
  These bounds, with $M=1$, also apply to multiplicative subgroups of $\mathbb
F^\times_p$, whose order is $O(\sqrt{p})$. We adapt the above energy bound to
larger subgroups and obtain new bounds on gaps between elements in cosets of
subgroups of order $\Omega(\sqrt{p})$.
