A Refined Sum-Product Estimate via Higher Energies

Let $A\subset \mathbb{R}$ be a finite set. Combining the multiplicative slope estimate of Rudnev–Stevens, Cushman’s higher-energy regularization, Shakan’s $d^{+}$–$d^{\times }$ decomposition, and Solymosi’s classical sum–product estimate, we prove $$\mathrm{\mid }AA{\mathrm{\mid }}^{204}\mathrm{\mid }A+A{\mathrm{\mid }}^{301}\gtrsim \mathrm{\mid }A{\mathrm{\mid }}^{675},$$ where $\gtrsim$ suppresses a fixed polylogarithmic factor in $\mathrm{\mid }A\mathrm{\mid }$. Consequently, for every $\epsilon >0$, $$\max \{\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }\}{\gg }_{\epsilon }\mathrm{\mid }A{\mathrm{\mid }}^{135\mathrm{/}101-\epsilon }\mathrm{.}$$ The proof is organized around two intermediate estimates. For every nonempty finite set $B\subset {\mathbb{R}}_{>0}$, $$d^{\times }(B)\mathrm{\mid }BB{\mathrm{\mid }}^{12}\mathrm{\mid }B+B{\mathrm{\mid }}^{16}\gtrsim \mathrm{\mid }B{\mathrm{\mid }}^{38},$$ whereas for every nonempty finite set $U\subset \mathbb{R}$, $$d^{+}(U{)}^{17}\mathrm{\mid }U+U{\mathrm{\mid }}^{29}\gtrsim \mathrm{\mid }U{\mathrm{\mid }}^{46}\mathrm{.}$$