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Arithmetic properties of sparse subsets ...

Potgieter, Paul...

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Arithmetic properties of sparse subsets of $\mathbb{Z}^n$ by Potgieter, Paul ( Author )

Content Provider	Lemonjar Open Access
Publisher	Australian National University
Publication Date	08-08-2023
Summary	Arithmetic progressions of length $3$ may be found in compact subsets of the reals that satisfy certain Fourier -- as well as Hausdorff -- dimensional requirements. It has been shown that a very similar result holds 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 $\mathbb{Z}^n$ satisfying appropriate density and Fourier-decay conditions.
ISBN	-
Item Type	Article
File Type	pdf
File Size	29.34 KB
Language	English
Subjects	-
Unlimited	MYR 0.01
Total read	-
URL	http://arxiv.org/abs/1602.01634

Citation

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AMA 11th EditionReferences

Potgieter, Paul. Arithmetic properties of sparse subsets of $\mathbb{Z}^n$. Australian National University. August 08, 2023. Accessed October 01, 2025. https://online.provdatabase.com/cite/12802

APA 7th EditionReferences

Potgieter, Paul. (2023). Arithmetic properties of sparse subsets of $\mathbb{Z}^n$. Australian National University. https://online.provdatabase.com/cite/12802

Chicago 17th EditionReferences

Potgieter, Paul. Arithmetic properties of sparse subsets of $\mathbb{Z}^n$. Australian National University. 2023. https://online.provdatabase.com/cite/12802

HarvardReferences

Potgieter, Paul. (2023). Arithmetic properties of sparse subsets of $\mathbb{Z}^n$. [Online]. Australian National University. Available from: https://online.provdatabase.com/cite/12802. (Accessed 01 October 2025).

Harvard AustraliaReferences

Potgieter, Paul, 2023, Arithmetic properties of sparse subsets of $\mathbb{Z}^n$, Australian National University, viewed 01 October 2025, <https://online.provdatabase.com/cite/12802>

MLA 9th EditionWorks Cited

Potgieter, Paul. Arithmetic properties of sparse subsets of $\mathbb{Z}^n$. Australian National University, 2023. PROV Database, https://online.provdatabase.com/cite/12802.

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