Discrete analogues in harmonic analysis

The study of discrete analogues in harmonic analysis originated in the early 1900's when M. Riesz noted that his work on the L p boundedness of the Hilbert transform implied the boundedness of a closely related discrete operator. Results for several other simple discrete analogues also follow trivially from their counterparts in the classical setting, but many discrete operators present distinctive difficulties apparently unapproachable from the continuous perspective. It was not until the work of J. Bourgain in the late 1980's, and in particular the introduction of ideas from the circle method of analytic number theory, that a renewed study of discrete analogues began in earnest.;This thesis presents new results for several classes of discrete operators. First, this thesis presents sharp results for families of discrete fractional integral operators along paraboloids. These results are further generalized to operators defined in terms of arbitrary positive definite quadratic forms with integer coefficients. Second, this thesis presents results on the boundedness of twisted discrete singular Radon transforms and related discrete oscillatory integral operators. Third, this thesis presents results for a discrete analogue of fractional integration on the Heisenberg group, as well as generalizations to fractional integral operators associated to symplectic bilinear forms of lower rank. The methods employed include classical analytic methods as well as techniques from number theory, including the circle method, theta functions, and exponential sums.