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A Host--Kra ${\mathbf F}_2^ω$-system of order $5$ that is not Abramov of order $5$, and non-measurability of the inverse theorem for the $U^6({\mathbf F}_2^n)$ norm

Asgar Jamneshan, Or Shalom, Terence Tao

Abstract

It was conjectured by Bergelson, Tao, and Ziegler \cite{btz} that every Host--Kra $\F_p^ω$-system of order $k$ is an Abramov system of order $k$. This conjecture has been verified for $k \leq p+1$. In this paper we show that the conjecture fails when $k=5, p=2$. We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $f: \F_2^n \to \C$ of large Gowers norm $\|f\|_{U^6(\F_2^n)}$ which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial $e(P)$, but with the property that all such phase polynomials $e(P)$ are ``non-measurable'' in the sense that they cannot be well approximated by functions of a bounded number of random translates of $f$. A simpler version of our construction can also be used to answer a question of Candela, González-Sánchez, and Szegedy \cite{CGSS}.

A Host--Kra ${\mathbf F}_2^ω$-system of order $5$ that is not Abramov of order $5$, and non-measurability of the inverse theorem for the $U^6({\mathbf F}_2^n)$ norm

Abstract

It was conjectured by Bergelson, Tao, and Ziegler \cite{btz} that every Host--Kra -system of order is an Abramov system of order . This conjecture has been verified for . In this paper we show that the conjecture fails when . We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function of large Gowers norm which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial , but with the property that all such phase polynomials are ``non-measurable'' in the sense that they cannot be well approximated by functions of a bounded number of random translates of . A simpler version of our construction can also be used to answer a question of Candela, González-Sánchez, and Szegedy \cite{CGSS}.
Paper Structure (17 sections, 29 theorems, 274 equations)

This paper contains 17 sections, 29 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Overview of proof
  3. Acknowledgments
  4. Notation
  5. A characterization of coboundaries on $\mathbb{F}_2^n$
  6. A non-trivial cocycle
  7. Two key nilspaces
  8. Counterexample to the strong inverse conjecture
  9. Constructing the counterexample
  10. Equidistribution theory for $Q,S$
  11. Concluding the argument
  12. Nilspaces, filtered abelian groups, and non-classical polynomials
  13. Nilspaces
  14. Filtered abelian groups
  15. Polynomials
  16. ...and 2 more sections

Key Result

Theorem 1.5

For any given choice of $k$ and $p$, Conjecture btz-conj implies Conjecture inv-conj-strong (and hence also Conjecture inv-conj).

Theorems & Definitions (85)

  • Conjecture 1.1: Inverse conjecture for the Gowers norm
  • Conjecture 1.2: Bergelson--Tao--Ziegler conjecture
  • Conjecture 1.3: Strong inverse conjecture for the Gowers norm
  • Example 1.4
  • Theorem 1.5: Application of correspondence principle
  • Theorem 1.6: Counterexample to strong inverse conjecture
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1: $2$-homogeneous cocycles on elementary abelian $2$-groups
  • ...and 75 more