A dot-product bound from separate growth-minimizing bases

Abstract

For every finite $P\subset {\mathbb{R}}^2$ we prove $|\{p\cdot q:p,q\in P\}|\gg |P{|}^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295=2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality $|F^{(2)}{G}^{(2)}/(F^{(2)}G)|\le |AB|(|AF|/|A|{)}^4(|BG|/|B|{)}^3$: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: ${\max }_{p\in P}|p\cdot P|=O(|P|/\sqrt{\log |P|})$ is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.