Exponentially Larger Affine and Projective Caps

TL;DR

This paper introduces a novel construction method for affine and projective caps in spaces over odd prime fields, achieving exponentially larger sizes for certain primes and significantly improving previous lower bounds.

Contribution

It presents a new construction technique for caps in all affine spaces with odd prime modulus, leading to exponential growth improvements for primes congruent to 5 mod 6.

Findings

Achieves exponential growth of caps in affine and projective spaces for primes p ≡ 5 mod 6

Improves lower bounds from approximately 8.09^n to nearly 9^n for certain primes

Provides explicit constructions for primes up to 41

Abstract

In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach (2017) and Ellenberg and Gijswijt (2017)), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus $p$. Moreover, we show that for all primes $p \equiv 5 \bmod 6$ with $p \leq 41$, the new construction leads to an exponentially larger growth of the affine and projective caps in $\mathrm{AG}(n,p)$ and $\mathrm{PG}(n,p)$. For example, when $p=23$, the existence of caps with growth $(8.0875\ldots)^n$ follows from a three-dimensional example of Bose (1947), and the only improvement had been to $(8.0901\ldots)^n$ by Edel (2004), based on a six-dimensional example. We improve this lower bound to $(9-o(1))^n$.