Gowers norms for singular measures

2013/08/13 by Marc Carnovale, Carnovale, Marc
Mathematics · #11B25 #28A78 #42A32 #42A38 #42A45 #42B10 #42B35 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1308.2721

openalex publication_date 2013/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Gowers introduced the notion of uniformity norm ‖f‖_U^k(G) of a bounded function f:G→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^∞(G)↦ L^∞ (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 \mathbbT^d, and similarly an object ‖μ‖_U^k. We provide criteria for \triangle^k μ to exist, which turns out to be equivalent to finiteness of ‖|μ|‖_U^k, and show that when μ is absolutely continuous with density f, then the objects which we have introduced are reduced to the standard \triangle^kf and ‖f‖_U^k(\mathbbT). 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.

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