Parseval-type identities for Gowers uniformity norms in finite abelian groups

TL;DR

This paper establishes Parseval-type identities and isometries for generalized Gowers uniformity norms in finite abelian groups, enhancing the understanding of their structure and orthogonality properties.

Contribution

It introduces new Parseval-type identities for Gowers norms, generalizing previous orthogonality relations in finite abelian groups.

Findings

Derived Parseval identities for Gowers norms

Established isometries for generalized Gowers norms

Enhanced understanding of orthogonality in Gowers inner products

Abstract

Orthogonality relations for cubes of characters in Gowers inner products $\langle \cdot \rangle_{d,l}$ lead to Parseval-type identities and isometries for suitably generalized Gowers uniformity norms $U^{d,l}$.