  The aim of this paper is to investigate the intersection problem between two
linear sets in the projective line over a finite field. In particular, we
analyze the intersection between two clubs with eventually different maximum
fields of linearity. Also, we analyze the intersection between the linear set
defined by the polynomial $\alpha x^{q^k}+\beta x$ and other linear sets having
the same rank; this family contains the linear set of pseudoregulus type
defined by $x^q$. The strategy relies on the study of certain algebraic curves
whose rational points describe the intersection of the two linear sets. Among
other geometric and algebraic tools, function field theory and the Hasse-Weil
bound play a crucial role. As an application, we give asymptotic results on
semifields of BEL-rank two.
