Sets of large values of correlation functions for polynomial cubic configurations
نویسنده
چکیده
We prove that for any set E ⊆ Z with upper Banach density d(E) > 0, the set “of cubic configurations” in E is large in the following sense: for any k ∈ N and any ε > 0, the set {(n1, . . . , nk) ∈ Z k : d( ⋂ e1,...,ek∈{0,1} (E − (e1n1 + · · ·+ eknk))) > d (E) k − ε} is an AVIP0set. We then generalize this result to the case “of polynomial cubic configurations” e1p1(n) + · · · + ekpk(n) where the polynomials pi:Z d −→ Z are assumed to be sufficiently algebraically independent.
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