Inverse theorem for certain directional Gowers uniformity norms

Abstract

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||2kU(H1,...,Hk) = E x?G,h1?H1,...,hk?Hk ? ?h1... ??hk f(x) where ??uf(x) : = f(x + u)?f(x) is the discrete multiplicative derivative. Suppose that G is a direct sum of subspaces G = U1 ? U2 ?...? Uk. In this paper we prove the inverse theorem for the norm ||?||U(U1,...,Uk,G,...,G), with ? copies of G in the subscript, which is the simplest interesting unknown case of the inverse problem for the directional Gowers uniformity norms. Namely, writing ||? ||U for the norm above, we show that if f : G ? C is a function bounded by 1 in magnitude and obeying ||f||U ? c, provided ? < p, one can find a polynomial ?: G ? Fp of degree at most k + ? ? 1 and functions gi : ?j?[k]\{i} Uj ? {z ? C: |z| ? 1} for i ? [k] such that |Ex?G f(x)??(x) ? i? [k] gi(x1,... ,xi?1, xi+1,..., xk)| ? (exp(Op,k,?(1))(Op,k,?(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.