“ Mean square limit for lattice points in a sphere ”

The three-dimensional case is the most difficult one. A version of Theorem 1.1 is known for a long time for the circle (see [Cra] and [Lan1]) and for the d-dimensional ball when d ≥ 4 (see [Wal]). In [Ble1] a similar statement was proved for any strictly convex (in the sense that the curvature of the boundary is positive everywhere) oval in the plane with the origin inside the oval. Making an analogy to the theory of renormalization group in statistical mechanics we may say that three is the critical dimension for the problem under consideration. Namely, the series of squared Fourier amplitudes of N(R) converges when d < 3 and diverges when d ≥ 3 (in fact logarithmically diverges when d = 3). The criticality of d = 3 is reflected then in the appearance of the log-correction in (1.1). It is easy to show that