A Refined Sum-Product Estimate via Higher Energies

Abstract

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 \[ |AA|^{204}|A+A|^{301}\gtrsim |A|^{675}, \] where $\gtrsim$ suppresses a fixed polylogarithmic factor in $|A|$. Consequently, for every $\varepsilon>0$, \[ \max\{|A+A|,|AA|\}\gg_\varepsilon |A|^{135/101-\varepsilon}. \] The proof is organized around two intermediate estimates. For every nonempty finite set $B\subset\mathbb{R}_{>0}$, \[ d^\times(B)|BB|^{12}|B+B|^{16}\gtrsim |B|^{38}, \] whereas for every nonempty finite set $U\subset\mathbb{R}$, \[ d^+(U)^{17}|U+U|^{29}\gtrsim |U|^{46}. \]