On equidistributed directions in finite affine planes
Abstract
We investigate a recently introduced generalisation of determined directions in affine planes. A direction (d) in an affine plane is a point of the line at infinity in the projective completion, and hence corresponds to a parallel class. A (multi)set S of points is equidistributed from direction (d) if all lines from that parallel class intersect S in the same number of points. In this paper, we give a construction of point (multi)sets that are inequidistributed from exactly 3 directions in any finite translation plane, and prove that all such (multi)sets arise from this construction, generalising a result of Kiss and Somlai in Desarguesian planes of prime order. We also use ideas from algebraic graph theory to prove results on directions associated with a pair of point sets S and T. If for every direction (d) either S or T is equidistributed from (d), we prove that |S ∩ T| = |S||T|/q2, where q is the order of the affine plane. In the appendix, we give a combinatorial alternative approach to proving this equality, which can be of independent interest. We also introduce the notion of directions cross-determined by a pair of sets S and T, and prove that |S| |T| ≤ q2 if (S,T) cross-determines at most half of the directions, where equality forces S = T.
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BibTeX
@misc{adriaensen2026equidistributed,
title = {{On equidistributed directions in finite affine planes}},
author = {Sam Adriaensen and Bence Csajbók and Zsuzsa Weiner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40023},
eprint = {2609.40023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40023},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On equidistributed directions in finite affine planes
AU - Sam Adriaensen
AU - Bence Csajbók
AU - Zsuzsa Weiner
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40023
UR - https://arxiv.org/abs/2609.40023
AB - We investigate a recently introduced generalisation of determined directions in affine planes. A direction (d) in an affine plane is a point of the line at infinity in the projective completion, and hence corresponds to a parallel class. A (multi)set S of points is equidistributed from direction (d) if all lines from that parallel class intersect S in the same number of points. In this paper, we give a construction of point (multi)sets that are inequidistributed from exactly 3 directions in any finite translation plane, and prove that all such (multi)sets arise from this construction, generalising a result of Kiss and Somlai in Desarguesian planes of prime order. We also use ideas from algebraic graph theory to prove results on directions associated with a pair of point sets S and T. If for every direction (d) either S or T is equidistributed from (d), we prove that |S ∩ T| = |S||T|/q^2, where q is the order of the affine plane. In the appendix, we give a combinatorial alternative approach to proving this equality, which can be of independent interest. We also introduce the notion of directions cross-determined by a pair of sets S and T, and prove that |S| |T| ≤ q^2 if (S,T) cross-determines at most half of the directions, where equality forces S = T.
ER -