Algebraic capsets
Cassie Grace, José Felipe Voloch
TL;DR
This work tackles the problem of constructing complete capsets in $\mathbb{F}_3^n$ that approach the known lower bounds, by developing algebraic-construction techniques over extensions $\mathbb{F}_{3^m}$. It introduces explicit configurations, notably unions of parabolas in $\mathbb{F}_{3^m}^2$ and paraboloids in $\mathbb{F}_q^3$, and a Frobenius-type framework to combine multiple parabolic constraints, yielding small complete capsets of size $O(\sqrt{3^n})$ for all $n$. The paper proves that a two-parabola construction is complete for odd $m$ and analyzes limitations for even $m$, providing computational data and a Frobenius-based path to even smaller complete capsets, while also giving a complete paraboloid construction $Q=\{(x,y,x^2-\lambda y^2)\}$ with $\lambda$ non-square. A key contribution is delivering the smallest known complete capsets with sizes matching the best known lower bounds, and addressing Csajbok’s Problem 5.1 for $p=3$ through concrete algebraic-geometric constructions and counting arguments. Overall, the results expand the toolkit for capset problems by linking algebraic geometry over finite fields to explicit complete-capset constructions.
Abstract
Capsets are subsets of $\mathbb{F}_3^n$ with no three points on a line and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of $\mathbb{F}_3$. In particular we construct the smallest known complete capsets with size proportional to the best known lower bound.