Abstract
Given a subset of the integers of zero density, we define the weaker notion of the fractional density of such a set. We show that a version of a theorem of Łaba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of the integers with fractional density whose characteristic functions have Fourier coefficients that decay sufficiently rapidly.
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Communicated by Yang Wang.
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Potgieter, P. Arithmetic Progressions in Salem-Type Subsets of the Integers. J Fourier Anal Appl 17, 1138–1151 (2011). https://doi.org/10.1007/s00041-011-9179-0
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DOI: https://doi.org/10.1007/s00041-011-9179-0
Keywords
- Arithmetic progressions
- Hausdorff dimension
- Fractional density
- Fourier coefficients
- Salem-type set
- Roth’s theorem
- Fejér kernel
- Varnavides’s theorem