  Motivated by the study of the growth rate of the number of geodesics in flat
surfaces with bounded lengths, we study generalizations of such problems for K3
surfaces. In one generalization, we give a result regarding the upper bound on
the asymptotics of the number of classes of irreducible special Lagrangians in
K3 surfaces with bounded period integrals. In another generalization, we give
the exact leading term in the asymptotics of the number of Mukai vectors of
semistable coherent sheaves on algebraic K3 surfaces with bounded central
charges, with respect to generic Bridgeland stability conditions.
