Periodic nilsequences and inverse theorems on cyclic groups

TL;DR

This paper provides an alternative proof for the periodic inverse theorem for Gowers norms, ensuring the associated nilsequence is strongly N-periodic, and introduces a new construction in the category of nilsequences and nilmanifolds.

Contribution

It offers a new proof that relies solely on the Green–Tao–Ziegler inverse theorem, removing a technical condition and introducing a novel construction in nilsequence theory.

Findings

Alternative proof of N-periodic inverse theorem using Green–Tao–Ziegler theorem

Removal of a technical condition in the periodic inverse theorem

Introduction of a new construction in nilsequences and nilmanifolds

Abstract

The inverse theorem for the Gowers norms, in the form proved by Green, Tao and Ziegler, applies to functions on an interval $[M]$. A recent paper of Candela and Sisask requires a stronger conclusion when applied to $N$-periodic functions; specifically, that the corresponding nilsequence should also be $N$-periodic in a strong sense. In most cases, this result is implied by work of Szegedy (and Camarena and Szegedy) on the inverse theorem. This deduction is given in Candela and Sisask's paper. Here, we give an alternative proof, which uses only the Green--Tao--Ziegler inverse theorem as a black box. The result is also marginally stronger, removing a technical condition from the statement. The proof centers around a general construction in the category of nilsequences and nilmanifolds, which is possibly of some independent interest.