Quadratic Expansion over Prime Fields via Centered Collisions and Popular-Sum Amplification

Abstract

Let $p$ be an odd prime, let $\varnothing \ne A\subseteq {\mathbb{F}}_p$ have cardinality $N$, and let $f\in {\mathbb{F}}_p[x,y]$ be a non-degenerate quadratic polynomial. Writing $S=|A+A|$ and $M=|f(A,A)|$, we prove the full-range trade-off $S^8{M}^6\gtrsim N^{17}(1+N^3/p^2{)}^{-3}$. Consequently, $\max \{|A+A|,|f(A,A)|\}\gtrsim \min \{N^{17/14},p^{3/7}{N}^{4/7}\}$, and in particular the exponent $17/14$ holds throughout $N\le p^{2/3}$. The proof combines a centered collision estimate for $F(u,v,w)=f(u+v,w)$, a mixed fourth-energy bound, and a popular-sum amplification. Two complementary incidence estimates enter the argument: a centered spectral bound in the dense collision regime and a point--plane bound in the sparse regime.

arXiv comment10 pages. No figures