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 (Formula Presented) 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 (Formula Presented) with ℓ 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 k· kU for the norm above, we show that if f: G → C is a function bounded by 1 in magnitude and obeying kfkU > c, provided ℓ < p, one can find a polynomial α: G → Fp of degree at most k + ℓ − 1 and functions gi: ⊕j2[k]r{i} Uj → {z ⊕ C: |z| 6 1} for i ⊕ [k] such that The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.