A refinement of Cauchy–Schwarz complexity

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