Classes and equivalence of linear sets in PG(1,q^n)

The equivalence problem of Fq-linear sets of rank n of PG(1,q^n) is investigated, also in terms of the associated variety, projecting configurations, Fq-linear blocking sets of Rédei type and MRD-codes. We call an Fq-linear set L_U of rank n in PG(W,Fq^n)=PG(1,q^n) simple if for any n-dimensional Fq-subspace V of W, L_V is PΓL(2,q^n)-equivalent to L_U only when U and V lie on the same orbit of ΓL(2,q^n). We prove that U={(x,Tr_{q^n/q}(x)):x∈Fq^n} defines a simple Fq-linear set for each n. We provide examples of non-simple linear sets not of pseudoregulus type for n>4 and we prove that all Fq-linear sets of rank 4 are simple in PG(1,q^4).