Uniformity norms, their weaker versions, and applications

TL;DR

This paper demonstrates that Gowers uniformity norms are essentially equivalent to simpler, weaker norms under certain conditions, enabling new applications like a variant of the Koopman--von Neumann decomposition and a proof of the inverse theorem.

Contribution

It establishes the equivalence between Gowers norms and weaker norms, facilitating easier analysis and new applications in additive combinatorics.

Findings

Gowers norms are equivalent to certain weaker norms under mild hypotheses.

A variant of the Koopman--von Neumann decomposition is developed.

A proof of the inverse theorem for Gowers norms using pseudorandomness conditions.

Abstract

We show that, under some mild hypotheses, the Gowers uniformity norms (both in the additive and in the hypergraph setting) are essentially equivalent to certain weaker norms which are easier to understand. We present two applications of this equivalence: a variant of the Koopman--von Neumann decomposition, and a proof of the relative inverse theorem for the Gowers $U^s[N]$-norm using a norm-type pseudorandomness condition.