Linear forms from the Gowers uniformity norm

TL;DR

This paper demonstrates how to replace the linear forms condition with a Gowers uniformity norm assumption to prove a relative Szemerédi theorem for arithmetic progressions, advancing methods in additive combinatorics.

Contribution

It introduces a new approach linking Gowers uniformity norms to linear forms conditions for relative Szemerédi theorems.

Findings

Replaces linear forms condition with Gowers uniformity norm assumption

Provides a new proof technique for relative Szemerédi theorem

Enhances understanding of uniformity norms in additive combinatorics

Abstract

This is a companion note to our paper 'A relative Szemer\'edi theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemer\'edi theorem for $(r+1)$-term arithmetic progressions assuming a linear forms condition. Here we show how to replace this condition with an assumption about the Gowers uniformity norm $U^r$.