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

TL;DR

This paper proves the Jamneshan-Tao conjecture for finite abelian groups with bounded rank by establishing an inverse theorem linking Gowers norms to nilsequences.

Contribution

It introduces an inverse theorem for 1-bounded functions on finite abelian groups of bounded rank, confirming the conjecture in this setting.

Findings

Confirmed the Jamneshan-Tao conjecture for groups of rank at most R

Proved that functions with non-trivial Gowers norm correlate with bounded complexity nilsequences

Established a new inverse theorem for 1-bounded functions on these groups

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.