Discrete analogues in harmonic analysis -: I:: l2 estimates for singular radon transforms

This paper studies the discrete analogues of singular Radon transforms. We prove the l(2) boundedness for those operators that are "quasi-translation-invariant" The approach used is related to the "circle-method" of Hardy and Littlewood, and requires multi-dimensional extensions of Weyl sums and Gauss sums, as well as variants that replace scalar sums by operator sums.