On the few products, many sums problem

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 $\Omega(\sqrt{p})$.