An upper bound for the Erd\H{o}s unit distance problem in the plane

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An upper bound for the Erd\H{o}s unit distance problem in the plane

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Author(s): Theophilus Agama ¹ ^,

Publication date (Electronic preprint): 9 February 2023

Journal: AfricArXiv Preprints

Publisher: ScienceOpen

Keywords: compression gap, compression estimates, mass of compression

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Abstract

In this paper, using the method of compression, we prove a stronger upper bound for the Erd\H{o}s unit distance problem in the plane by showing that $\begin{align}\# \bigg\{||\vec{x_j}-\vec{x_t}||:\vec{x}_t, \vec{x_j}\in \mathbb{E}\subset \mathbb{R}^2,~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq n\bigg\}\ll_2 n^{1+o(1)}.\nonumber\end{align}$

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Title: AfricArXiv Preprints

Publisher: ScienceOpen

Publication date (Electronic preprint): 9 February 2023

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[1 ] African Institute for mathematical sciences;

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[* ]Email: theophilus@ 123456aims.edu.gh .

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Theophilus Agama https://orcid.org/0000-0001-7790-9368

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DOI: 10.14293/111.000/000056.v1

SO-VID: 778a4af1-df9a-4cdc-9c54-47216826b5d3

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This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

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Date received : 9 February 2023

An upper bound for the Erd\H{o}s unit distance problem in the plane

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