J Fourier Anal Appl https://doi.org/10.1007/s00041-018-9615-5 Bohr Sets in Triple Products of Large Sets in Amenable Groups 1 2 Michael Björklund · John T. Griesmer Received: 28 March 2017 / Revised: 3 April 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We answer a question of Hegyvári and Ruzsa concerning effective estimates of the Bohr-regularity of certain triple sums of sets with positive upper Banach densities in the integers. Our proof also works for any discrete amenable group, and it does not require all addends in the triple products we consider to have positive (left) upper Banach densities; one of the addends is allowed to only have positive upper asymptotic density with respect to a (possibly very sparse) ergodic sequence. Keywords Bohr sets · Densities · Measurable recurrence 1 Main Results Let Γ be a discrete group. If (W, σ) is a ﬁnite-dimensional unitary Γ-representation, we deﬁne the (possibly trivial) semi-norm | · | on Γ by |γ| = sup σ(γ)w − w : w = 1 ,for γ ∈ Γ, W W Communicated by Hans G. Feichtinger. B John T. Griesmer firstname.lastname@example.org Michael Björklund email@example.com Department of Mathematics, Chalmers, Gothenburg, Sweden Department of Applied Mathematics
Journal of Fourier Analysis and Applications – Springer Journals
Published: May 31, 2018
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