On the stability of sets of even type

A stability theorem says that a nearly extremal object can be obtained from an extremal one by “small changes”. In this paper, we prove a sharp stability theorem of sets of even type in PG(2,q)PG(2,q), q   even. As a consequence, we improve Blokhuis and Bruen's stability theorem on hyperovals and also on the minimum number of lines intersecting a point set of size at most q+2⌊q⌋−2; furthermore we improve on the lower bound for untouchable sets.