A NOTE ON THE BILINEAR BOGOLYUBOV THEOREM: TRANSVERSE AND BILINEAR SETS

摘要

A set P subset of F-p(n) x F-p(n), is called bilinear when it is the zero set of a family of linear and bilinear forms and transverse when it is stable under vertical and horizontal sums. A theorem of the first author provides a generalization of Bogolyubov's theorem to the bilinear setting. Roughly speaking, it implies that any dense transverse set P subset of F-p(n) x F-p(n), contains a large bilinear set. In this paper, we elucidate the extent to which a transverse set is forced to be (and not only contain) a bilinear set.