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Quadratic distances in even dimensions over prime fields
Abstract
Let p be an odd prime, let m≥1 be an integer, and let Q be a nondegenerate quadratic form on 𝔽p2m. For a nonempty set E⊆𝔽p2m, write ΔQ(E)={Q(x−y):x,y∈ E}. We prove that, whenever |E|>pm, |ΔQ(E)|≫ min{p/LE, |E|/(pm−1LE2)}, LE=log(2+(pm+1)/|E|), with an absolute implied constant independent of p, m and Q. This confirms the Erdős--Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case m=1, for small sets, we also prove that |ΔQ(E)|≫|E|/log(2|E|), (1≤ |E|≤ p), which is also optimal up to a logarithmic factor.
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BibTeX
@misc{pham2026quadratic,
title = {{Quadratic distances in even dimensions over prime fields}},
author = {Thang Pham and Chun-Yen Shen and Dung The Tran and Boqing Xue},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01795},
eprint = {2610.01795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01795},
}
RIS
TY - UNPB
TI - Quadratic distances in even dimensions over prime fields
AU - Thang Pham
AU - Chun-Yen Shen
AU - Dung The Tran
AU - Boqing Xue
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.01795
UR - https://arxiv.org/abs/2610.01795
AB - Let p be an odd prime, let m≥1 be an integer, and let Q be a nondegenerate quadratic form on 𝔽_p^2m. For a nonempty set E⊆𝔽_p^2m, write Δ_Q(E)={Q(x−y):x,y∈ E}. We prove that, whenever |E|>p^m, |Δ_Q(E)|≫ min{p/L_E, |E|/(p^m−1L_E^2)}, L_E=log(2+(p^m+1)/|E|), with an absolute implied constant independent of p, m and Q. This confirms the Erdős--Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case m=1, for small sets, we also prove that |Δ_Q(E)|≫|E|/log(2|E|), (1≤ |E|≤ p), which is also optimal up to a logarithmic factor.
ER -