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Abstract Let$$p_{1},\ldots ,p_{n}$$ be a set of points in the unit square and let$$T_{1},\ldots ,T_{n}$$ be a set of$$\delta $$ -tubes such that$$T_{j}$$ passes through$$p_{j}$$ . We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points$$p_{1},\ldots , p_{n} \in [0,1]^{2}$$ along with a line$$\ell _{j}$$ through each point$$p_{j}$$ , there exist$$j\neq k$$ for which$$d(p_{j}, \ell _{k}) \lesssim n^{-2/3+o(1)}$$ . It follows from the latter result that any set of$$n$$ points in the unit square contains three points forming a triangle of area at most$$n^{-7/6+o(1)}$$ . This new upper bound for Heilbronn’s triangle problem attains the high-low limit established in our previous work arXiv:2305.18253.
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Almost Sharp Bounds on the Number of Discrete Chains in the Plane
The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ=(δ₁,… ,δ_k) of k distances, a (k+1)-tuple (p₁,… ,p_{k+1}) of distinct points in ℝ^d is called a (k,δ)-chain if ‖p_j-p_{j+1}‖ = δ_j for every 1 ≤ j ≤ k. What is the maximum number C_k^d(n) of (k,δ)-chains in a set of n points in ℝ^d, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3.
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- Award ID(s):
- 1900460
- PAR ID:
- 10168550
- Date Published:
- Journal Name:
- Proceedings of 36th International Symposium on Computational Geometry (SoCG 2020)
- Volume:
- 164
- Issue:
- 48
- Page Range / eLocation ID:
- 1-15
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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