PreprintCC BY 4.0arXiv
A superlinear improvement on line-free sets in 𝔽p3
Abstract
Building on an earlier result of the author together with Elsholtz, Führer, Füredi, Pach, Simon and Velich, we present an improved construction for a line-free set in 𝔽p3, showing that rp(𝔽p3)≥ (p−1)3+1/8 p3/2 − O(p) as p→ ∞. This results in the first superlinear-term improvement over the standard hypercube construction {0,1,…,p−2}3. Via complementation, a line-free set in 𝔽p3 corresponds to a $2$-blocking set in the affine geometry AG(3,p), hence we also obtain an upper bound of 3p2−1/8p3/2+O(p) on the smallest size of such a $2$-blocking set.
Related
- Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets
- Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets
- Point sets determining few angles are almost contained in a line or circle
- Coding multiplicity in self-similar sets on the line
- No-three-in-line sets on the checkerboard grid
BibTeX
@misc{kovacs2026superlinear,
title = {{A superlinear improvement on line-free sets in $\mathbbF_p^3$}},
author = {Benedek Kovács},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.23437},
eprint = {2605.23437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.23437},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A superlinear improvement on line-free sets in 𝔽_p^3
AU - Benedek Kovács
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.23437
UR - https://arxiv.org/abs/2605.23437
AB - Building on an earlier result of the author together with Elsholtz, Führer, Füredi, Pach, Simon and Velich, we present an improved construction for a line-free set in 𝔽_p^3, showing that r_p(𝔽_p^3)≥ (p−1)^3+1/8 p^3/2 − O(p) as p→ ∞. This results in the first superlinear-term improvement over the standard hypercube construction {0,1,…,p−2}^3. Via complementation, a line-free set in 𝔽_p^3 corresponds to a $2$-blocking set in the affine geometry AG(3,p), hence we also obtain an upper bound of 3p^2−1/8p^3/2+O(p) on the smallest size of such a $2$-blocking set.
ER -