  In 2005, Li, Tao, Thiele and the author raised a general question concerning
upper bounds for a class of multilinear oscillatory integral operators, and
established such bounds in a few cases. Most cases remain open. The present
paper is concerned with sublevel set bounds, which would be a consequence of
the oscillatory integral bounds, if valid. These sublevel set bounds are
established here in a weak form but in nearly full generality, subject only to
a rationality hypothesis. The proof relies on an extension of Szemeredi's
theorem due to Furstenberg and Katznelson.
