We prove a version of Furstenberg's ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X, B, mu, T), integers 0 less than or equal to j < k, and E subset of X with mu(E) > 0, we show that there exists n equivalent to j (mod k) with mu(E boolean AND T(-n) E boolean AND T(-2n) E boolean AND T(-3n)E) > 0, so long as T(k) is ergodic. This result requires a deeper understanding of the limit of some nonconventional ergodic averages and the introduction of a new class of systems, the 'Quasi-Affine Systems'.