  Motivated by recent experimental data on dichloro-tetrakis thiourea-nickel
(DTN) [T.A. Soldatov $\textit{et al}$, Phys. Rev. B ${\bf 101}$, 104410
(2020)], a model of antiferromagnet on a tetragonal lattice with single-ion
easy-plane anisotropy in the tilted external magnetic field is considered.
Using the smallness of the in-plane field component, we analytically address
field dependence of the energy gap in ``acoustic'' magnon mode and transverse
uniform magnetic susceptibility in the ordered phase. It is shown that the
former is non-monotonic due to quantum fluctuations, which was indeed observed
experimentally. The latter is essentially dependent on the ``optical'' magnon
rate of decay on two magnons. At magnetic fields close to the one which
corresponds to the center of the ordered phase, it leads to experimentally
observed dynamical diamagnetism phenomenon.
