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

  • Thang Pham EMAIL logo
Veröffentlicht/Copyright: 26. März 2024
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 37 Heft 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.

References

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Received: 2023-12-21
Published Online: 2024-03-26
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
https://doi.org/10.1515/forum-2023-0470
Schlagwörter für diesen Artikel
Triangle; simplex; finite fields; incidences

Artikel in diesem Heft

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
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