  We consider properties of zero and near-zero fermionic modes in lattice
gluodynamics. The modes are known to be sensitive to the topology of the
underlying gluonic fields in the quantum vacuum state of the gluodynamics. We
find evidence that these modes are fine tuned, that is exhibit sensitivity to
both physical (one can say, hadronic) scale and to the ultraviolet cutoff.
Namely, the density of the states is in physical units while the localization
volume of the modes tends to zero in physical units with the lattice spacing
tending to zero. We discuss briefly possible theoretical implications and also
include some general, review-type remarks.
