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Structural theorems on the distance sets over finite fields

  • Doowon Koh , Minh Quy Pham and Thang Pham EMAIL logo
Published/Copyright: May 25, 2023

Abstract

Let F q be a finite field of order 𝑞. Iosevich and Rudnev [ErdƑs distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007), 12, 6127–6142] proved that, for any set A ⊂ F q d , if | A | ≫ q d + 1 2 , then the distance set Δ ⁹ ( A ) contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be d 2 in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture is still wide open in higher dimensions. To fill the gap, we need to understand more about the structures of the distance sets; the main purpose of this paper is to provide some structural theorems on the distribution of square and non-square distances.

MSC 2010: 52C10; 11T23

Award Identifier / Grant number: NRF-2018R1D1A1B07044469

Award Identifier / Grant number: 101.99-2021.09

Funding statement: D. Koh was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MIST) (No. NRF-2018R1D1A1B07044469). T. Pham was supported by the National Foundation for Science and Technology Development (NAFOSTED) Project 101.99-2021.09 (Title: ErdƑs–Falconer distance conjecture over finite fields).

Acknowledgements

T. Pham would like to thank the VIASM for the hospitality and for the excellent working conditions.

  1. Communicated by: Christopher D. Sogge

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Received: 2022-05-03
Revised: 2022-12-04
Published Online: 2023-05-25
Published in Print: 2023-07-01

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