2025/05/09 by Michael Björklund, Björklund, Michael, Tobias Hartnick +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2505.05980
openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group G, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over G together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or Θ-) transforms and Marklof-Strömbergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis. Our main applications concern dynamical systems (X, μ) in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced G-representations into Lp(X, μ) for appropriate values of p. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case. In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schrödinger representations into the L2-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms.