Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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NOTE ON THE PINNED DISTANCE PROBLEM OVER FINITE FIELDS

  • Koh, Doowon (Department of Mathematics Chungbuk National University)
  • Received : 2022.06.08
  • Accepted : 2022.08.24
  • Published : 2022.08.15
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Abstract

Let 𝔽q be a finite field with odd q elements. In this article, we prove that if E ⊆ 𝔽dq, d ≥ 2, and |E| ≥ q, then there exists a set Y ⊆ 𝔽dq with |Y| ~ qd such that for all y ∈ Y, the number of distances between the point y and the set E is ~ q. As a corollary, we obtain that for each set E ⊆ 𝔽dq with |E| ≥ q, there exists a set Y ⊆ 𝔽dq with |Y| ~ qd so that any set E ∪ {y} with y ∈ Y determines a positive proportion of all possible distances. The averaging argument and the pigeonhole principle play a crucial role in proving our results.

Keywords

Acknowledgement

This work was conducted during the research year of Chungbuk National University in 2022, and was supported by Basic Science Research Programs through National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07044469).

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