An Inverse Theorem for the Uniformity Seminorms Associated with the Action of \mathbb F^∞_p

2009/01/31 by Vitaly Bergelson, Terence Tao, Tamar Ziegler · 2 citations
Computer Science · Mathematics · #Abelian group #Action (physics) #Analytic Number Theory Research #Ergodic theory #Finite field #Finite group #Inverse #Limits and Structures in Graph Theory #Norm (philosophy) #Polynomial and algebraic computation #Structured program theorem #math.CO #math.DS #msc:37A35

paper · pdf · doi:10.1007/s00039-010-0051-1

published as Geom. Funct. Anal. 19 (2010), No. 6, 1539-1596 · 59 pages, 2 figures, to appear, GAFA. Referee suggestions incorporated. Also, the paper has been shortened at the request of the journal; previous versions on the arXiv can thus be viewed as extended versions

arxiv created 2009/06/08 · openalex publication_date 2010/02/15 · arxiv updated 2012/01/04 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05

Abstract

Let \mathbb F a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action (T_g)_g ∈ \mathbb F^ω of the infinite abelian group \mathbb F^ω on a probability space X = (X, \mathcal B, μ) is generated by phase polynomials φ : X → S¹ of degree less than C(k) on X, where C(k) depends only on k. In the case where k ≤ \rm char(\mathbb F) we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ≤ \rm char(\mathbb F) , with a partial result in low characteristic.

Citations

• On sets of integers containing k elements in arithmetic progression
• The Descriptive Set Theory of Polish Group Actions
• Universal characteristic factors and Furstenberg averages
• The primes contain arbitrarily long arithmetic progressions
• A new proof of Szemerédi's theorem

Cited by

• AN INVERSE THEOREM FOR THE GOWERSU4-NORM
• \bigoplus_{p∈ P}\mathbbF_p-Systems as Abramov Systems

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