Gowers norms for singular measures

TL;DR

This paper extends Gowers norms to singular measures on the torus, establishing criteria for their existence, and introduces a higher-order inner product with a Gowers-Cauchy-Schwarz inequality.

Contribution

It develops an analogue of Gowers norms for singular measures, providing criteria for their existence and connecting them to classical norms for absolutely continuous measures.

Findings

Criteria for the existence of the measure analogue of $ riangle^k$

Reduction to classical Gowers norms for absolutely continuous measures

Introduction of a higher-order inner product with Gowers-Cauchy-Schwarz inequality

Abstract

Gowers introduced the notion of uniformity norm $\|f\|_{U^k(G)}$ of a bounded function $f:G\rightarrow\mathbb{R}$ on an abelian group $G$ in order to provide a Fourier-theoretic proof of Szemeredi's Theorem, that is, that a subset of the integers of positive upper density contains arbitrarily long arithmetic progressions. Since then, Gowers norms have found a number of other uses, both within and outside of Additive Combinatorics. The $U^k$ norm is defined in terms of an operator $\triangle^k : L^{\infty}(G)\mapsto L^{\infty} (G^{k+1})$. In this paper, we introduce an analogue of the object $\triangle^k f$ when $f$ is a singular measure on the torus $\mathbb{T}^d$, and similarly an object $\|\mu\|_{U^k}$. We provide criteria for $\triangle^k \mu$ to exist, which turns out to be equivalent to finiteness of $\||\mu|\|_{U^k}$, and show that when $\mu$ is absolutely continuous with density $f$, then the objects which we have introduced are reduced to the standard $\triangle^kf$ and $\|f\|_{U^k(\mathbb{T})}$. We further introduce a higher-order inner product between measures of finite $U^k$ norm and prove a Gowers-Cauchy-Schwarz inequality for this inner product.