On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In \cite{DoDu2008}, the intersection problem for subgeometries of $\PG(n,q)$ is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline $\PG(1,q)$ with a linear set in $\PG(1,q^h)$ and investigate the existence of {\em irregular} sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd $q$, on the size of the intersection of two different linear sets of rank 3 in $\PG(1,q^h)$.