Abstract
We describe a series of results on the boundary of harmonic analysis, ergodic and analytic number theory. The central objects of study are maximal averages taken over integer points of varieties defined by integral polynomials. The mapping properties of such operators are intimately connected with those of certain exponential sums studied in analytic number theory. They can be used to prove pointwise ergodic results associated to singular averages defined in terms of polynomials both in the commutative and non-commutative settings.
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