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PreprintCC BY 4.0arXiv

Congruence Classes in 𝔽p2: Sharp Results via an Energy Approach

  • Thang Pham
  • Le Quang Ham
  • Steven Senger
  • Dung The Tran
  • Boqing Xue
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Abstract

Let Tk(E) denote the set of congruence classes of ordered k-tuples of pairwise distinct points of E. Let p≡3pmod4 be prime. For E⊂𝔽p2 with 3≤|E|≤ p3/4, we prove that |T3(E)|≫|E|11/6; for 4≤|E|≤ p3/4, we prove that |T4(E)|≫|E|3/log|E|. For every fixed k≥5 and k≤|E|≤ p3(k−2)/(3k−4), we prove that |Tk(E)|≫k|E|k−1. The proofs proceed by bounding the moments of the overlap function of rigid motions. The main inputs are an exact identity involving the distance energy and an incidence bound obtained by viewing rigid motions as lines over 𝔽p(i). The bound for k=4 is sharp up to a logarithmic factor, while the bounds for k≥5 are sharp up to constants.

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Cite

BibTeXRIS
BibTeX
@misc{pham2026congruence,
  title = {{Congruence Classes in $\mathbbF_p^2$: Sharp Results via an Energy Approach}},
  author = {Thang Pham and Le Quang Ham and Steven Senger and Dung The Tran and Boqing Xue},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29784},
  eprint = {2609.29784},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29784},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Congruence Classes in 𝔽_p^2: Sharp Results via an Energy Approach
AU  - Thang Pham
AU  - Le Quang Ham
AU  - Steven Senger
AU  - Dung The Tran
AU  - Boqing Xue
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29784
UR  - https://arxiv.org/abs/2609.29784
AB  - Let T_k(E) denote the set of congruence classes of ordered k-tuples of pairwise distinct points of E. Let p≡3pmod4 be prime. For E⊂𝔽_p^2 with 3≤|E|≤ p^3/4, we prove that |T_3(E)|≫|E|^11/6; for 4≤|E|≤ p^3/4, we prove that |T_4(E)|≫|E|^3/log|E|. For every fixed k≥5 and k≤|E|≤ p^3(k−2)/(3k−4), we prove that |T_k(E)|≫_k|E|^k−1. The proofs proceed by bounding the moments of the overlap function of rigid motions. The main inputs are an exact identity involving the distance energy and an incidence bound obtained by viewing rigid motions as lines over 𝔽_p(i). The bound for k=4 is sharp up to a logarithmic factor, while the bounds for k≥5 are sharp up to constants.
ER  -