A refinement of Cauchy–Schwarz complexity

Abstract

We introduce a notion of complexity for systems of linear forms called sequential Cauchy–Schwarz complexity, which is parametrized by two positive integers k,ℓ and refines the notion of Cauchy–Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy–Schwarz complexity at most (k,ℓ) then any average of 1-bounded functions over this system is controlled by the 21−ℓ-th power of the Gowers Uk+1-norms of the functions. For ℓ=1 this agrees with Cauchy–Schwarz complexity, but for ℓ>1 there are systems that have sequential Cauchy–Schwarz complexity at most (k,ℓ) whereas their Cauchy–Schwarz complexity is greater than k. Our main application illustrates this with systems over a prime field Fp, denoted by Φk,M, which can be viewed as M-dimensional arithmetic progressions of length k. For each M≥2 we prove that Φk,M has sequential Cauchy–Schwarz complexity at most (k−2,|Φk,M|) (where |Φk,M| is the number of forms in the system), whereas the Cauchy–Schwarz complexity of Φk,M can be greater than k−2. Thus we obtain polynomial true-complexity bounds for Φk,M with exponent 2−|Φk,M|. A recent general theorem of Manners, proved independently with different methods, implies a similar application but with different polynomial true-complexity bounds, as explained in the paper. In separate work, we use our application to give a new proof of the inverse theorem for Gowers norms on Fpn, and related results on ergodic actions of Fpω.