Classifications and constructions of minimum size linear sets

This paper aims to study linear sets of minimum size in the projective line, that is $\mathbb{F}_q$-linear sets of rank $k$ in $\mathrm{PG}(1,q^n)$ admitting one point of weight one and having size $q^{k-1}+1$. Examples of these linear sets have been found by Lunardon and the second author (2000) and, more recently, by Jena and Van de Voorde (2021). However, classification results for minimum size linear sets are known only for $k\leq 5$. In this paper we provide classification results for those $L_U$ admitting two points with complementary weights. We construct new examples and also study the related $\mathrm{\Gamma L}(2,q^n)$-equivalence issue. These results solve an open problem posed by Jena and Van de Voorde. The main tool relies on two results by Bachoc, Serra and Z\'emor (2017 and 2018) on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs.