Orthogonal projections and incidence bounds in planes over prime order fields

Let $p$ be an odd prime and let $E\subset {\mathbb{F}}_p^2$ with $\mathrm{\mid }E\mathrm{\mid }=p^a$, where $0<a\le 1$. For a direction $V$ (a $1$-dimensional subspace of ${\mathbb{F}}_p^2$), let ${\pi }^V:{\mathbb{F}}_p^2\to {\mathbb{F}}_p^2\mathrm{/}V$ 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 $a\mathrm{/}2\le s\le a$, define $$T_s^{1,2}(E):=\{V\in G(1,{\mathbb{F}}_p^2):\mathrm{\mid }{\pi }^V(E)\mathrm{\mid }<p^s\}\mathrm{.}$$ We prove $$\mathrm{\mid }{T}_s^{1,2}(E)\mathrm{\mid }\ll \min \{p^{\frac{5}{2}s-a},p^{6s-3a},p^s\},$$ which improves the best previously known estimates over prime fields in the range $a\mathrm{/}2\le s<2a\mathrm{/}3$, 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.