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The Jamneshan-Tao conjecture for finite abelian groups of bounded rank

Pablo Candela, Diego González-Sánchez, Balázs Szegedy

TL;DR

The paper establishes the Jamneshan–Tao inverse theorem for finite abelian groups of bounded rank, showing that any 1-bounded function with large $U^{k+1}$-norm correlates with a degree-$k$ nilsequence on a nilmanifold of bounded complexity after passing to a bounded-index subgroup. A key innovation is the reduction to quasitoral nilspaces, which decompose into a bounded number of toral components, enabling correlation with nilsequences on a subgroup. The authors then solve the nilsequence-extension problem from a subgroup to the full group, addressing split and non-split cases via a lifting construction and a multivariable extension theorem, while preserving complexity. The results extend higher-order Fourier analysis on finite groups, generalizing prior inverse theorems (e.g., Green–Tao–Ziegler, Bergelson–Tao–Ziegler) to the bounded-rank setting and advancing understanding of JT in this regime.

Abstract

We confirm the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed integer $R$ (i.e. finite abelian groups generated by at most $R$ elements), by proving an inverse theorem for 1-bounded functions of non-trivial Gowers norm on such groups, concluding that such a function must correlate non-trivially with a nilsequence of bounded complexity.

The Jamneshan-Tao conjecture for finite abelian groups of bounded rank

TL;DR

The paper establishes the Jamneshan–Tao inverse theorem for finite abelian groups of bounded rank, showing that any 1-bounded function with large -norm correlates with a degree- nilsequence on a nilmanifold of bounded complexity after passing to a bounded-index subgroup. A key innovation is the reduction to quasitoral nilspaces, which decompose into a bounded number of toral components, enabling correlation with nilsequences on a subgroup. The authors then solve the nilsequence-extension problem from a subgroup to the full group, addressing split and non-split cases via a lifting construction and a multivariable extension theorem, while preserving complexity. The results extend higher-order Fourier analysis on finite groups, generalizing prior inverse theorems (e.g., Green–Tao–Ziegler, Bergelson–Tao–Ziegler) to the bounded-rank setting and advancing understanding of JT in this regime.

Abstract

We confirm the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed integer (i.e. finite abelian groups generated by at most elements), by proving an inverse theorem for 1-bounded functions of non-trivial Gowers norm on such groups, concluding that such a function must correlate non-trivially with a nilsequence of bounded complexity.
Paper Structure (5 sections, 21 theorems, 17 equations)

This paper contains 5 sections, 21 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Background on nilspaces
  3. The role of quasitoral nilspaces in the bounded-rank setting
  4. Extending nilsequences
  5. Proof of the inverse theorem

Key Result

Theorem 1.1

For any $k,R\in \mathbb{N}$ and $\delta>0$, there is $\varepsilon>0$ and a finite collection $\mathcal{N}_{k,R,\delta}$ of degree-$k$ filtered nilmanifolds $G/\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that the following hol

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3: Quasitoral nilspaces
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 45 more