Points below a parabola in affine planes of prime order

Technical Summary: Points Below a Parabola in Affine Planes of Prime Order

Problem Statement
This paper investigates the geometric and combinatorial properties of the set of points lying below a parabola in the Desarguesian affine plane $AG(2, p)$, where $p$ is a prime number. Specifically, the authors analyze the set $S = \{(x, y) \in \mathbb{F}_p^2 \mid y < f(x)\}$, where $f(x) = \alpha x^2 + \beta x + \gamma$ is a quadratic polynomial with $\alpha \neq 0$.

The study is motivated by recent work of Kiss and Somlai [KS24], who examined the set of points below the diagonal line ($y < x$) in $AG(2, p)$. They demonstrated that this set determines exactly three "special" directions (directions where lines do not intersect the set in an equidistributed manner) and that this set possesses a unique non-trivial automorphism. The authors of this paper seek to determine if similar phenomena occur for parabolic sets, despite the fact that parabolas generally possess less symmetry than lines.

Methodology
The authors employ a combination of finite geometry, number theory, and combinatorial analysis:

1. Ordering and Projection: Utilizing the canonical correspondence between elements of $\mathbb{F}_p$ and integers in $[0, p-1]$, the authors define an ordering on the field. They introduce the projection function $pr_{S,d}$, which maps a direction $d$ to the number of points in $S$ intersected by lines of slope $d$.
2. Quadratic Character Analysis: The core of the analysis relies on the quadratic character $\chi$ on $\mathbb{F}_p$. The authors utilize the Pólya-Vinogradov inequality (Result 2.1) to bound character sums over intervals and results by Sárközy and Sokolovskiĭ (Result 2.2) regarding the lower bounds of these sums.
3. Algebraic Transformation: To analyze the distribution of intersection numbers, the authors apply linear transformations to the variables in the defining inequality of $S$. This allows them to relate the projection functions of different directions to one another.
4. Symmetry Group Analysis: The authors investigate the stabilizer of the set $S$ within the projective linear group $PGL(3, p)$ by distinguishing specific points at infinity based on the distribution of intersection numbers of lines passing through them.

Key Contributions and Results

• Cyclic Symmetry of Projections (Theorem 3.1):
  The primary structural result is that for any non-zero direction $d \in \mathbb{F}_p^*$, the projection function $pr_{S,d}$ is a cyclic shift of the projection function $pr_{S,1}$. Specifically, the values taken by the projection function for any non-vertical, non-horizontal direction are identical up to a shift in the domain. This implies that the set $S$ "looks the same" from all directions except for the vertical ($\infty$) and horizontal ($0$) directions.
  Furthermore, the image of these projection functions forms an interval of integers with a length bounded between $\frac{\sqrt{p}}{2\pi}$ and $\sqrt{p} \ln(p)$.

• Size of the Set (Theorem 3.3):
  The authors establish tight bounds on the cardinality of $S$. They prove that $|S|$ is always a multiple of $p$. Furthermore, the size of $S$ is close to $\binom{p}{2}$, with the deviation bounded by $c_p p \sqrt{p} \ln(p)$, where the constant $c_p$ depends on $p \pmod 4$ and $p \pmod 8$. In the specific case where $p \equiv 1 \pmod 4$ and $\gamma = 0$, the size is exactly $\binom{p}{2}$.

• Rigidity of Symmetries (Theorem 3.5):
  Contrary to the intuition that such uniform projection behavior might imply high symmetry, the authors prove that for sufficiently large $p$ ($p \geq 97$), the set $S$ has a unique non-trivial automorphism in $PGL(3, p)$. This automorphism is the reflection across the vertical axis: $(X, Y, Z) \mapsto (-X, Y, Z)$.
  The proof demonstrates that the directions $(0)$ and $(\infty)$ are uniquely distinguishable from all other directions based on the number of distinct intersection counts of lines passing through them. This rigidity contrasts with the "uniform" appearance of the set from other directions.

• Classification of Isomorphism Classes (Remark 3.7):
  The authors classify the number of non-isomorphic sets of this type. By considering affine transformations and the complement of the sets, they determine that there are exactly $3p+1$ pairwise non-isomorphic sets formed by points below or on a parabola (including variations with $\leq, >, \geq$).

Significance
The paper highlights a counter-intuitive phenomenon in finite geometry: a set of points can exhibit a high degree of "uniformity" in its projection functions (appearing identical from almost all directions) while possessing very limited geometric symmetry (only one non-trivial automorphism).

This work extends the results of Kiss and Somlai [KS24] from linear boundaries to quadratic boundaries. It demonstrates that the property of having few special directions and cyclically shifted projection functions is not exclusive to sets defined by linear inequalities. The results rely heavily on the distribution of quadratic residues and character sums, linking the geometric properties of the point set directly to the arithmetic properties of the underlying finite field.

The authors note that these findings are part of a broader effort to generalize the results of [KS24], with an accompanying paper [ASW24] exploring connections to codes generated by incidence matrices of finite projective planes.