Distribution of lattice points in the shifted balls

In this work we study the mean value of the difference between the number of integer points and the volume of a ball as a function of the center of a ball in the unit cube [0, 1]3, applying new method. This mean value is estimated by its possible exact value. Using methods of Fourier analysis, we lead the question to the estimates of double trigonometric integrals. This method allows consider the question on lattice points in domains of arbitrary nature without any symmetry.