On the distribution of lattice points on spheres and level surfaces of polynomials

The irregularities of distribution of lattice points on spheres and on level surfaces of polynomials are measured in terms of the discrepancy with respect to caps. It is found that the discrepancy depends on diophantine properties of the direction of the cap. If the direction of the cap is diophantine, in case of the spheres, close to optimal upper bounds are found. The estimates are based on a precise description of the Fourier transform of the set of lattice points on polynomial surfaces.