Renitent lines

TL;DR

This paper studies special point sets in finite affine planes where most lines in certain parallel classes intersect the set uniformly, with a focus on lines called renitent that exhibit exceptional intersection behavior.

Contribution

It introduces the concept of renitent lines in finite affine planes and analyzes their regularity and intersection properties.

Findings

Renitent lines exhibit predictable intersection patterns.

Most lines in a parallel class intersect the set uniformly.

Results depend on the characteristic of the finite field.

Abstract

There are many examples for point sets in finite geometry, which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod $p$, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behaviour of these renitent lines.