Journal of Integer Sequences, Vol. 24 (2021), Article 21.4.4

Arithmetic Properties of Sparse Subsets of Zn

Paul Potgieter
Department of Decision Sciences
University of South Africa
P. O. Box 392
Pretoria 0003
South Africa


Arithmetic progressions of length three may be found in compact subsets of the reals that satisfy certain Fourier- as well as Hausdorff-dimensional requirements. Similar results hold in the integers under analogous conditions, with Fourier dimension being replaced by the decay of a discrete Fourier transform. In this paper we make this correspondence more precise, using a well-known construction by Salem. Specifically, we show that a subset of the integers can be mapped to a compact subset of the continuum in a way which preserves certain dimensional properties as well as arithmetic progressions of arbitrary length. The higher-dimensional version of this construction is then used to show that certain parallelogram configurations must exist in sparse subsets of Zn satisfying appropriate density and Fourier-decay conditions.

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Received August 11 2020; revised version received August 12 2020; April 15 2021. Published in Journal of Integer Sequences, April 16 2021.

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