On the number of lattice points in three-dimensional solids of revolution

We derive an accurate estimate for the order of magnitude of the remainder term in the problem of the number of lattice points in families of homothetic domains belonging to the class of three-dimensional solids of revolution with smooth boundaries (under certain additional conditions). This estimate is realized in the case of the solid bounded by a standardly embedded torus, for which the second term of the expansion, which describes the dependence of the number of lattice points on the dilation parameter, is written in explicit form.