  U(1) gauge theory with the Villain action on a cubic lattice approximation of
three- and four-dimensional torus is considered. The naturally chosen
correlation functions converge to the correlation functions of the R-gauge
electrodynamics on three- and four-dimensional torus as the lattice spacing
approaches zero only for the special scaling. This special scaling depends on a
choice of a correlation function system. Another scalings give the degenerate
continuum limits. The Wilson criterion for the confinement is ambiguous. The
asymptotics of the smeared Wilson loop integral for the large loop perimeters
is defined by the density of the loop smearing over a torus which is
transversal to the loop plane. When the initial torus radius tends to infinity
the correlation functions converge to the correlation functions of the R-gauge
Euclidean electrodynamics.
