PreprintCC BY 4.0arXiv
Geometric progressions in distance sets in odd dimensions over prime fields
Abstract
For E⊂𝔽pd, let Δ(E)={Qd(x−y):x,y∈ E}, where Qd(z)=z12+⋯+zd2. We introduce and study the problem of the existence of nondegenerate three-term geometric progressions in the distance set Δ(E) when d is odd. If MGP,d(p) denotes the largest size of a set whose distance set contains no such progression, then, for every prime p≥7 and every integer m≥0, pmMGP,1(p)≤ MGP,2m+1(p) ≤4pmMGP,1(p). For all sufficiently large primes p, we also prove pm+7/26≤ MGP,2m+1(p) ≤ Cpm+16/25, where C is an absolute positive constant.
Related
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BibTeX
@misc{pham2026geometric,
title = {{Geometric progressions in distance sets in odd dimensions over prime fields}},
author = {Thang Pham and Dung The Tran and Boqing Xue},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34758},
eprint = {2610.34758},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34758},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric progressions in distance sets in odd dimensions over prime fields
AU - Thang Pham
AU - Dung The Tran
AU - Boqing Xue
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34758
UR - https://arxiv.org/abs/2610.34758
AB - For E⊂𝔽_p^d, let Δ(E)={Q_d(x−y):x,y∈ E}, where Q_d(z)=z_1^2+⋯+z_d^2. We introduce and study the problem of the existence of nondegenerate three-term geometric progressions in the distance set Δ(E) when d is odd. If M_GP,d(p) denotes the largest size of a set whose distance set contains no such progression, then, for every prime p≥7 and every integer m≥0, p^mM_GP,1(p)≤ M_GP,2m+1(p) ≤4p^mM_GP,1(p). For all sufficiently large primes p, we also prove p^m+7/26≤ M_GP,2m+1(p) ≤ Cp^m+16/25, where C is an absolute positive constant.
ER -