  We consider discrete analogues of fractional Radon transforms involving
integration over paraboloids defined by positive definite quadratic forms. We
prove that such discrete operators extend to bounded operators from $\ell^p$ to
$\ell^q$ for a certain family of kernels. The method involves an intricate
spectral decomposition according to major and minor arcs, motivated by ideas
from the circle method of Hardy and Littlewood. Techniques from harmonic
analysis, in particular Fourier transform methods and oscillatory integrals, as
well as the number theoretic structure of quadratic forms, exponential sums,
and theta functions, play key roles in the proof.
