Discrete analogues in harmonic analysis, I: ℓ 2 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.