Arithmetic Properties of Sparse Subsets of
Department of Decision Sciences
University of South Africa
P. O. Box 392
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.
Full version: pdf,
Received August 11 2020; revised version received August 12 2020; April 15 2021.
Published in Journal of Integer Sequences,
April 16 2021.
Journal of Integer Sequences home page