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We use Grothendieck theorem to prove a structure theorem for multicorrelationsequences of length two, associated with two (not necessarily commuting)measure preserving actions on a probability space. We use this to deduce amultiple recurrence result concerning products of linear terms, and a partitionregularity result of certain systems of quadratic equations, building on thework of Frantzikinakis and Host.

An application of Grothendieck theorem to the theory of multicorrelation  sequences, multiple recurrence and partition regularity of quadratic  equations