Orthogonal projections and incidence bounds in planes over prime order fields

TL;DR

The paper studies orthogonal projections in the plane over a prime field ${\mathbb F}_p^2$ and aims to bound the number of exceptional projections of a finite point set $E$. It introduces two key incidence bounds: one for point–line incidences when the lines lie in few directions, and another for Cartesian-product configurations $A\times B$ against lines of the form $C\times C$ controlled by additive energy $E^+(C)$. These incidence results are then used to derive a new bound on the number of small projections $|T_s^{1,2}(E)|$, advancing toward Chen’s conjecture for planar finite-field settings. The approach hinges on projective duality, a refined half-product incidence bound (Stevens–de Zeeuw framework with pruning), and energy-based arguments for structured line families, yielding improved estimates in various parameter ranges. Overall, the work strengthens the link between incidence geometry and projection questions in finite fields and provides concrete bounds that sharpen prior results in the two-dimensional prime-field case.

Abstract

We give an upper bound on the number of exceptional orthogonal projections of a small set of points in a plane over a prime order field, which makes progress toward a conjecture of Chen (2018). This theorem relies on a new upper bound on the number of incidences between an arbitrary set of points and a set of lines pointing in a reasonably small number of distinct directions. We also prove a new upper bound on the number of incidences between points and lines when both are Cartesian products in an appropriate sense.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2: Stevens, de Zeeuw
• Lemma 2.3
• proof