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arXiv · 2608.10355

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

Zhi Yao
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Abstract

Let $p$ be an odd prime, let $\varnothing\neq A\subseteq\mathbb F_p$ 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 $S^8M^6\gtrsim N^{17}(1+N^3/p^2)^{-3}$. 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.

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Zhi Yao. 2026-08-11. Quadratic Expansion over Prime Fields via Centered Collisions and Popular-Sum Amplification. https://arxiv.org/abs/2608.10355

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