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Source. Theorem 1.1, p. 1, 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. The proof, through Theorem 1.4 (p. 2) and Theorem 1.9 (p. 6), was read for structure only and is not checked here.

Statement

A $\delta$-tube is the $\delta$-neighborhood of a line, and a tube $T$ passes through a point $p$ when its central line $\ell_T\subset T$ passes through $p$ (p. 1).

Theorem 1.1 (p. 1, quoted). "For all $\varepsilon>0$ the following holds for $\delta<\delta_0(\varepsilon)$. Let $p_1,\ldots,p_n$ be a set of points in $[0,1]^2$ along with a $\delta$-tube $T_j$ through each point. If $n\geqslant\delta^{-3/2-\varepsilon}$, there is some nontrivial incidence $p_j\in T_k$ where $j\ne k$."

The paper states that Corollary 1.2 (p. 1), the version with lines and distances, is equivalent to this theorem; see Corollary 1.2. By subsampling it also gives the incidence count of Corollary 1.3.

Proof pointer

The paper remarks (p. 2) that Theorem 1.1 is the case $s=0$ of Theorem 1.4, which is proved in §5.2 from the phase-space incidence bound Theorem 1.9.

Dependencies

Theorems 1.4 and 1.9 of the same paper.

Bears on

• Problem 507: through Corollary 1.2 this theorem gives the paper's Theorem 1.8, a triangle of area at most $n^{-7/6+o(1)}$ among any $n$ points of the unit square; see Theorem 1.8. The theorem itself is about points and tubes, not triangles.