The minimum size of a linear set

TL;DR

This paper determines the minimum size of linear sets in projective spaces over finite fields, providing new bounds and confirming a conjecture for specific cases.

Contribution

It establishes the minimum size of Fq-linear sets in PG(1, q^n) and derives bounds for PG(2, q^n), confirming Sziklai's conjecture for k=n.

Findings

Minimum size of linear sets in PG(1, q^n) determined

Lower bounds on points in linear sets in PG(2, q^n) established

Confirmation of Sziklai's conjecture for k=n

Abstract

In this paper, we first determine the minimum possible size of an Fq-linear set of rank k in PG(1, q^n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an Fq- linear set of rank k in PG(2, q^n). In the case k = n, this confirms a conjecture by Sziklai in [9].