Inverse theorem for certain directional gowers uniformity norms

Let G be a finite-dimensional vector space over a prime field Fp with some subspaces H1, ... , Hk. Let f : G-+ C be a function. Generalizing the notion of Gowers uniformity norms, Austin introduced directional Gowers uniformity norms of f over (H1, ... , Hk) as ⠍ ⠍ ⠍f ⠍2k U(H & iota;,...,Hk) = E xEG,h & iota;EH & iota;,...,hkEHk & BULL;& UDelta;h & iota;... & BULL;& UDelta;hkf(x) where & BULL;& UDelta;uf (x) : = f(x + u)f (x) is the discrete multiplicative derivative. Suppose that G is a direct sum of subspaces G = U1 & REG; U2 & REG; & BULL; & BULL; & BULL; & REG; Uk. In this paper we prove the inverse theorem for the norm II & BULL; IIU(U & iota;,...,Uk,G,...,G), with l copies of G in the subscript, which is the simplest interesting un-known case of the inverse problem for the directional Gowers uniformity norms. Namely, writing II & BULL; IIU for the norm above, we show that if f : G-+ C is a func-tion bounded by 1 in magnitude and obeying IIf IIU c, provided l < p, one can find a polynomial & alpha;: G-+ Fp of degree at most k + l - 1 and functions gi: & REG;jE[k].{i} Uj-+ {z E C: |z| 1} for i E [k] such that ⠌⠌⠌⠌ExEGf(x)& omega;& alpha;(x)Hgi(x1, . . . ,xi-1, xi+1, . . ., xk) ⠌⠌⠌⠌ iE[k] > ( exp(Op,k,l(1))(Op,k,l(c-1)))-1. The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.