Sets of large values of correlation functions for polynomial cubic configurations

We prove that for any set E subset of Z with upper Banach density d(*) (E) > 0, the set 'of cubic configurations' in E is large in the following sense: for any k is an element of N and any epsilon > 0, the set {(n1 , . . . , nk) is an element of Z(k) ; d(*) (boolean AND(e1 , . . . , ek is an element of {0,1}) (E - (e(1)n(1) + . . . + e(k)n(k))) ) > d(*) (E)(2k) - epsilon} is an AVIP(0)(*)-set. We then generalize this result to the case of ` polynomial cubic configurations' e(1) p(1) (n) + . . . + e(k) p(k) (n), where the polynomials p(i) : Z(d) -> Z are assumed to be sufficiently algebraically independent.