A non-flag arithmetic regularity lemma and counting lemma

TL;DR

This paper extends Green and Tao's arithmetic regularity and counting lemmas to all systems of linear forms, removing the previous flag condition restriction.

Contribution

It introduces a new arithmetic regularity and counting lemma that applies universally to all systems of linear forms, broadening the scope of prior results.

Findings

Applicable to all systems of linear forms

Generalizes previous lemmas without flag condition

Enables broader applications in additive combinatorics

Abstract

Green and Tao's arithmetic regularity lemma and counting lemma together apply to systems of linear forms which satisfy a particular algebraic criterion known as the `flag condition'. We give an arithmetic regularity lemma and counting lemma which applies to all systems of linear forms.