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Non-existence of sets with few special directions
Abstract
Let p be an odd prime and let S⊆𝔽p2 have cardinality divisible by p. We prove that, for all sufficiently large primes p, no such set has exactly four special directions, and obtain a conditional extension to larger numbers of special directions under an affine-independence assumption on the corresponding projection functions. A separate second-moment argument shows more generally that, for every fixed k≥4, no subset of 𝔽p2 has exactly k special directions once p is sufficiently large. In fact, the result holds uniformly for k up to a positive constant times √p/log p. In contrast, for multisets every prescribed collection of at most p directions can occur as the set of special directions of a {0,1,2}-valued multiset on 𝔽p2.
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BibTeX
@misc{ghidelli2026nonexistence,
title = {{Non-existence of sets with few special directions}},
author = {Luca Ghidelli and Gergely Kiss and Gábor Somlai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21779},
eprint = {2609.21779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21779},
}
RIS
TY - UNPB
TI - Non-existence of sets with few special directions
AU - Luca Ghidelli
AU - Gergely Kiss
AU - Gábor Somlai
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21779
UR - https://arxiv.org/abs/2609.21779
AB - Let p be an odd prime and let S⊆𝔽_p^2 have cardinality divisible by p. We prove that, for all sufficiently large primes p, no such set has exactly four special directions, and obtain a conditional extension to larger numbers of special directions under an affine-independence assumption on the corresponding projection functions. A separate second-moment argument shows more generally that, for every fixed k≥4, no subset of 𝔽_p^2 has exactly k special directions once p is sufficiently large. In fact, the result holds uniformly for k up to a positive constant times √p/log p. In contrast, for multisets every prescribed collection of at most p directions can occur as the set of special directions of a {0,1,2}-valued multiset on 𝔽_p^2.
ER -