Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces

Thang Pham

doi:10.1515/forum-2023-0470

Article

Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces

Thang Pham

Published/Copyright: March 26, 2024

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From the journal Forum Mathematicum Volume 37 Issue 2

Abstract

The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in 𝔽 q 2 . More precisely, for A , B , C ⊂ 𝔽 q 2 , if | A | ⁢ | B | ⁢ | C | 1 2 ≫ q 4 , then for any λ ∈ 𝔽 q ∖ { 0 } , the number of congruence classes of triangles with vertices in A × B × C and one side-length λ is at least ≫ q 2 . In higher dimensions, we obtain similar results for k-simplex but under a slightly stronger condition. Compared to the well-known L 2 method in the literature, our approach offers better results in both conditions and conclusions. When A = B = C , the second goal of this paper is to give a new and unified proof of the best current results on the distribution of simplex due to Bennett, Hart, Iosevich, Pakianathan and Rudnev (2017) and McDonald (2020). The third goal of this paper is to study a Furstenberg-type problem associated to a set of rigid motions. The main ingredients in our proofs are incidence bounds between points and rigid motions. While the incidence bounds for large sets are due to the author and Semin Yoo (2023), the bound for small sets will be proved by using a point–line incidence bound in 𝔽 q 3 due to Kollár (2015).

Keywords: Triangle; simplex; finite fields; incidences

MSC 2020: 52C10; 51E25; 51E30

Communicated by Jan Bruinier

Acknowledgements

Thang Pham would like to thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) for the hospitality and for the excellent working condition.

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Received: 2023-12-21

Published Online: 2024-03-26

Published in Print: 2025-02-01

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Articles in the same Issue

https://doi.org/10.1515/forum-2023-0470

Keywords for this article

Triangle; simplex; finite fields; incidences