Orthogonal projections and incidence bounds in planes over prime order fields

Abstract

Let $p$ be an odd prime and let $E\subset \mathbb{F}_p^2$ with $|E|=p^a$, where $0<a\le 1$. For a direction $V$ (a $1$-dimensional subspace of $\mathbb{F}_p^2$), let $\pi^V:\mathbb{F}_p^2\to \mathbb{F}_p^2/V$ denote the quotient map. We bound the size of the exceptional set of directions for which the projection $\pi^V(E)$ is small. More precisely, for $a/2\le s\le a$, define $$ T_s^{1,2}(E):=\{V\in G(1,\mathbb{F}_p^2):\ |\pi^V(E)|<p^s\}. $$ We prove $$ |T_s^{1,2}(E)|\ll \min\{\,p^{\frac52 s-a},\ p^{6s-3a},\ p^s\,\}, $$ which improves the best previously known estimates over prime fields in the range $a/2\le s<2a/3$, and yields the first substantial progress toward Chen's 2018 conjecture. The key new ingredient is a novel point-line incidence bound, of independent interest, that yields a power saving when the line set spans only moderately many distinct directions. In the reverse direction, we also obtain an incidence estimate for Cartesian products $A\times B$ with line families $\{y=ax+b:\ a,b\in C\}$ with explicit dependence on the additive energy $E^+(C)$. We also discuss connections to the sum-set problem and the distinct dot-product values conjecture.