2021/03/10 by Milićević, Luka
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2103.06354
Let G be a finite-dimensional vector space over a prime field \mathbbFp with some subspaces H1, …, Hk. Let f \colon 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‖U(H1, …, Hk)2k = 𝔼x ∈ G,h1 ∈ H1, …, hk ∈ Hk ∂h1 … ∂hk f(x) where ∂u f(x) \colon= 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, \smash[b]\underbrace\scriptstyle G, …, G\scriptscriptstyle ℓ), 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 \colon G → ℂ is a function bounded by 1 in magnitude and obeying ‖f‖U ≥ c, provided ℓ < p, one can find a polynomial α\colon G → \mathbbFp of degree at most k + ℓ - 1 and functions gi \colon ⊕_j ∈ [k] ∖ \i\ Gj → \z ∈ ℂ \colon |z| ≤ 1\ for i ∈ [k] such that |𝔼x ∈ G f(x) ωα(x) ∏i ∈ [k] gi(x1, …, xi-1, xi+1, …, xk)| ≥ (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.