Authors: Milićević, Luka 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Journal: Publications de l'Institut Mathematique
Volume: 113
Issue: 127
First page: 1
Last page: 56
Issue Date: 2023
Rank: M24
ISSN: 0350-1302
DOI: 10.2298/PIM2327001M
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.
Keywords: bias | directional uniformity norms | partition rank
Publisher: Mathematical Institute of the Serbian Academy of Sciences and Arts

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