Direction-complete finite-field sets and quadratic secant lifting

Definitions and mathematical statement

Let $q$ be a prime power and let $n\geq1$. A direction in the vector space $\mathbb F_q^n$ is a one-dimensional linear subspace $\langle v\rangle=\{tv:t\in\mathbb F_q\}$ generated by a nonzero vector $v$. Two nonzero vectors specify the same direction exactly when they are nonzero scalar multiples. There are $(q^n-1)/(q-1)$ such directions. A subset $E\subseteq\mathbb F_q^n$ determines a direction $L$ if it contains distinct points $x,y$ such that $x-y\in L\setminus\{0\}$. The affine line through $x,y$ then has direction $L$.

Determine the extremal number \[ D(q,n)=\min\{|E|:E\subseteq\mathbb F_q^n, \text{ every direction is determined by }E\}. \] The field order and dimension are explicit parameters; one can seek exact values or sharp asymptotic bounds in specified regimes. Each unordered pair of distinct points determines only one direction, so every admissible set satisfies $\binom{|E|}{2}\geq(q^n-1)/(q-1)$. The question is how close configurations can come to this basic counting constraint, and what additional incidence restrictions prevent equality. The requirement is existence of a pair in every direction, not containment of a full affine line in every direction. In particular this is a difference-direction problem rather than the finite-field Kakeya problem. Both translations and invertible linear coordinate changes preserve admissibility and cardinality.

Short English statement

What is the smallest subset of a finite vector space whose point differences realize every direction?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001121 (AIM-COMBINATORICS-0246), 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 3.9. Original workshop question underlying AIM-COMBINATORICS-0246. https://aimath.org/WWN/additivecomb/additivecomb.pdf.