Finite-field point-line incidences and a projective no-wrap criterion

Definitions and mathematical statement

Let $p$ be a prime, let $P\subseteq\mathbb F_p^2$ be a set of distinct points, and let $\mathcal L$ be a set of distinct affine lines. A line is a set $\{(x,y):ax+by=c\}$ with $(a,b)\ne(0,0)$, with scalar multiples of an equation representing the same line. Write \[ I(P,\mathcal L)=|\{(v,\ell)\in P\times\mathcal L:v\in\ell\}|, \qquad n=|P|,\quad l=|\mathcal L|. \] The source asks for a prime-field analogue of the real point--line incidence bound and singles out logarithmically small configurations. Its unrestricted proposed bound has a large-configuration obstruction, so the precise remaining target here is the following.

For every fixed pair of constants $0<c\leq C<\infty$, do there exist $K(c,C)$ and $p_0(c,C)$ such that every prime $p\geq p_0$ and every configuration with \[ c\log p\leq n,l\leq C\log p \] satisfies \[ I(P,\mathcal L)\leq K(c,C)(nl)^{2/3}? \] The logarithm is natural, and the constant must be independent of the prime and the point and line choices. In this size regime the usual additive terms $n+l$ can be absorbed in the displayed estimate. No general-position assumptions are imposed: many points may be collinear, and many lines may pass through a point. Only prime fields are considered; extension fields introduce additional subfield configurations.

Short English statement

Does the real-plane incidence exponent remain valid over a prime field when both the point set and line set have logarithmic size?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001122 (AIM-COMBINATORICS-0247), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 3.10. Original workshop question underlying AIM-COMBINATORICS-0247. https://aimath.org/WWN/additivecomb/additivecomb.pdf.