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Source. Theorem 1.8, p. 4, of Alex Cohen, Cosmin Pohoata and Dmitrii Zakharov, Lower bounds for incidences, Invent. Math. 240 (2025), no. 3, 1045-1118, arXiv:2409.07658; read in arXiv:2409.07658v2 (18 March 2025), the edition named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page, and its short proof from Corollary 1.2 (p. 4) was read in full. Corollary 1.2 itself is not checked here.

Statement

Theorem 1.8 (p. 4, quoted). "For any $\varepsilon>0$, every set of $n$ points in the unit square contains a triangle of area $\Delta\lesssim_\varepsilon n^{-7/6+\varepsilon}$."

A triangle here has its three vertices among the given points. In the notation of §1.2 (p. 3), where $\Delta(n)$ is the smallest number such that any $n$ points of the unit square contain three forming a triangle of area at most $\Delta(n)$, the theorem says $\Delta(n)\le n^{-7/6+o(1)}$, as the abstract states. It improves the authors' earlier bound $\Delta\le n^{-8/7-1/2000}$ (Theorem 1.6, p. 3, from arXiv:2305.18253), and the exponent $7/6$ is the barrier of the high-low method that the paper describes on p. 4.

Proof pointer

p. 4. Pigeonholing on a grid of side $5/\sqrt n$ gives two points at distance at most $10/\sqrt n$; removing them and repeating gives $\lfloor n/4\rfloor$ disjoint pairs $(p_j,p_j')$ with $d(p_j,p_j')\le20/\sqrt n$. With $\ell_j$ the line through $p_j,p_j'$, Corollary 1.2 gives $j\ne k$ with $d(p_j,\ell_k)\lesssim_\varepsilon n^{-2/3+\varepsilon}$, and the triangle $p_k,p_k',p_j$ has area $\lesssim_\varepsilon n^{-7/6+\varepsilon}$.

Dependencies

Corollary 1.2 of the same paper.

Bears on

• Problem 507: the problem asks for the order of $\alpha(n)$ for the unit disk; this theorem is stated for the unit square. The transfer to the disk, $\alpha(n)\le4\Delta(n)$, and the resulting upper bound $\alpha(n)\ll n^{-7/6+o(1)}$ are recorded on the claim page. The theorem gives no lower bound and does not settle the order of $\alpha(n)$.