On the minimum size of linear sets

TL;DR

This paper generalizes a known lower bound on the size of linear sets in projective spaces, providing tight bounds for certain cases and constructions that attain these bounds.

Contribution

It extends the lower bound to linear sets intersecting subspaces in a canonical subgeometry and provides explicit constructions achieving equality.

Findings

Established a tight lower bound for linear sets meeting a subspace in a canonical subgeometry.

Provided constructions of linear sets that attain the bound.

Generalized previous bounds to broader configurations.

Abstract

Recently, a lower bound was established on the size of linear sets in projective spaces, that intersect a hyperplane in a canonical subgeometry. There are several constructions showing that this bound is tight. In this paper, we generalize this bound to linear sets meeting some subspace $\pi$ in a canonical subgeometry. We obtain a tight lower bound on the size of any $\mathbb F_q$-linear set spanning $\text{PG}(d,q^n)$ in case that $n \leq q$ and $n$ is prime. We also give constructions of linear sets attaining equality in the former bound, both in the case that $\pi$ is a hyperplane, and in the case that $\pi$ is a lower dimensional subspace.