On the number of directions formed by Cartesian products in $\mathbb{F}_{p^2}^2$

Technical Summary: On the Number of Directions Formed by Cartesian Products in $\mathbb{F}_{p^2}$

Problem Statement
The paper investigates the number of directions determined by a point set $U$ in the affine plane over a finite field $\mathbb{F}_q$, specifically focusing on Cartesian products $A \times A$ where $A \subset \mathbb{F}_{p^2}$ and $q = p^2$. A direction is defined by the slope of the line connecting two distinct points in the set.

While it is trivial that a set with more than $q$ points determines all possible directions, the problem becomes non-trivial for sets of size $|U| \le q$. In prime fields, sharp lower bounds are known, but over general finite fields, structured sets (particularly those concentrated on affine copies of subfields) can determine significantly fewer directions. The specific obstruction addressed here is when $A$ is contained within an affine copy of the proper subfield $\mathbb{F}_p$. In such cases, $A \times A$ determines at most $p$ directions, which is far below the quadratic scale expected for sets of size close to $p$.

The central question is: If $A$ is not contained in any affine copy of $\mathbb{F}_p$, what is the lower bound on the number of directions $|D(A \times A)|$?

Methodology
The proof synthesizes tools from algebraic combinatorics, finite geometry, and additive number theory:

1. Rédei Polynomial Method: The paper utilizes the Rédei polynomial $R(X, Y)$ associated with the point set. The roots of the specialized polynomial $R(X, y)$ correspond to the intercepts of lines with slope $y$ intersecting the set. The multiplicity of these roots relates to the number of points on each line.
2. Algebraic Multiplicity Parameters: The authors employ definitions from Fancsali, Sziklai, and Takács regarding two key parameters:
   • $s(y)$: The greatest power of $p$ such that every line of direction $y$ intersects the set in a number of points divisible by $s(y)$.
   • $t(y)$: An algebraic parameter derived from the auxiliary polynomial $H(X, y)$ in the factorization $R(X, Y)Q(X, Y) = X^q + H(X, Y)$. Specifically, $t(y)$ is the maximal power of $p$ such that $H(X, y)$ lies in $\mathbb{F}_q[X^{t(y)}]$ but not in $\mathbb{F}_q[X^{p \cdot t(y)}]$.
3. Structural Results on Direction Sets: The proof relies on a structural lemma (Lemma 2.7, implicit in Li and Roche-Newton) stating that if a set of directions $D(X)$ is closed under addition ($1+D(X) \subset D(X)$) and multiplication ($X D(X) \subset D(X)$), then $D(X)$ must be a subfield generated by $X$.
4. Additive Combinatorics: The argument uses variants of the Plünnecke-Ruzsa inequality (Lemmas 2.5 and 2.6) and a result by Li and Roche-Newton (Lemma 2.4) to bound the size of sumsets and product sets. These tools are used to derive contradictions if the set of directions fails to exhibit the necessary closure properties.

Key Contributions and Results
The main result, Theorem 1.1, establishes a quadratic lower bound for the number of directions determined by $A \times A$ in $\mathbb{F}_{p^2}$, provided $A$ avoids subfield structure.

• Theorem 1.1: Let $q = p^2$ for an odd prime $p$, and let $A \subset \mathbb{F}_q$. There exists an absolute constant $k > 1$ such that if $k \cdot p^{2/3} < |A| < p$ and $A$ is not contained in any affine copy of $\mathbb{F}_p$ (i.e., $A \not\subset c\mathbb{F}_p + d$ for any $c \in \mathbb{F}_q^*, d \in \mathbb{F}_q$), then:
  $$|D(A)| \ge \frac{|A|^2 + 1}{2}$$

Proof Logic:
The proof proceeds by normalizing $A$ such that $0, 1 \in A$. It then considers two cases:

1. Closure Case: If for all directions $r \in D(A)$, the set $|rA - A| \le p$, the authors use additive combinatorial bounds to show that $D(A)$ must be closed under addition and multiplication. By Lemma 2.7, this implies $D(A)$ is a subfield. Since $A \not\subset \text{affine copy of } \mathbb{F}_p$, $D(A)$ cannot be $\mathbb{F}_p$. If $D(A) = \mathbb{F}_{p^2}$, the bound holds trivially.
2. Non-Closure Case: If there exists a direction $y \in D(A)$ such that $|yA - A| > p$, the authors prove (Claim 3.1) that the algebraic multiplicity parameter $t(y)$ and the geometric parameter $s(y)$ both equal 1. Applying the lower bound theorem of Fancsali, Sziklai, and Takács (Theorem 2.3) with $t=1$ yields the desired quadratic bound $|D(A)| \ge (|A|^2 + 1)/2$.

Significance and Scope
The paper claims to resolve the lower bound problem for Cartesian products in the specific range $p^{2/3} < |A| < p$ (up to a constant factor), excluding the degenerate subfield case.

• Refinement of Existing Bounds: The work refines the general lower bounds obtained by Fancsali, Sziklai, and Takács for general point sets by explicitly handling the Cartesian product structure and the specific geometry of $\mathbb{F}_{p^2}$.
• Explicit Multiplicity: A key technical step is demonstrating that unless the set of directions forces a subfield structure, one can explicitly identify a direction where the algebraic multiplicity parameter $t$ is 1, allowing for the application of sharp bounds.
• Limitations: The authors note that extending this quadratic lower bound to the range $|A| < p^{2/3}$ remains an open problem. In that lower range, the best known bounds are sub-quadratic and rely on incidence geometry methods (citing [6] and [7]). The paper does not claim to solve the problem for all set sizes, nor does it provide a geometric characterization of sets attaining the optimal bound for $|A| < q$ in the general case, noting this remains unknown.