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Source. Theorem 1.4, p. 2, with the definitions on p. 2, 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 and the definitions it uses were read clause by clause on the printed page. The proof (§5.2, pp. 31-34) was read for structure only and is not checked here.

Statement

Definitions (p. 2). Lines are measured by $d(\ell_1,\ell_2)=|d(0,\ell_1)-d(0,\ell_2)|+|\theta(\ell_1)-\theta(\ell_2)|$, where $d(0,\ell)$ is the distance from the origin and $\theta(\ell)\in\mathbb R/\pi\mathbb Z$ the angle; the paper restricts attention to lines with $d(0,\ell)\le10$. A set $L$ of lines is a $(\delta,s,C)$-set if it is $\delta$-separated in this metric and $|L\cap B_w(\ell_0)|\le Cw^s|L|$ for every $w$-ball $B_w(\ell_0)$ with $w\in[\delta,1]$. The paper uses the same notion for sets of $\delta$-tubes through a point.

Theorem 1.4 (p. 2, quoted). "Fix $t\in[1,2]$ and $s\in[0,1]$ such that $2t+s>3$. There exists $\eta(t,s)>0$ such that the following holds for all $\delta<\delta_0(t,s)$. Let $P\subset[0,1]^2$ be a set of $\delta^{-t}$ many points. For each $p\in P$ let $\mathbb{T}_p$ be a $(\delta,s,\delta^{-\eta})$-set of $\delta$-tubes through $p$ and let $\mathbb{T}=\bigsqcup_p\mathbb{T}_p$. Then there is some nontrivial incidence between $P$ and $\mathbb{T}$, meaning there is a point $p\in P$ and a tube $T\in\mathbb{T}\setminus\mathbb{T}_p$ so that $p\in T$."

No separation or regularity is assumed of the point set $P$; the regularity hypothesis is on the tubes through each point. The case $s=0$ gives Theorem 1.1 (p. 2). The paper calls this result its main consequence (p. 7).

Proof pointer

§5.2 (pp. 31-34, by contradiction). Lift the point-tube pairs to the phase space $\Omega=[-1,1]^3$ of point-line pairs. When $s+t>2$ the Lipschitz property of the branching function already gives a contradiction (p. 32). When $s+t\le2$, choose the $u\times uw\times w$ phase-space rectangle maximizing $|\mathbf X\cap\mathbf R|u^{-\alpha}w^{-\beta}$ and blow up into it, so that the blown-up set is $(\alpha,\beta)$-Frostman, then apply Theorem 1.9 (the outline is on p. 6).

Dependencies

Theorem 1.9, Lemmas 3.6, 3.8, 3.11 and A.3 of the same paper.

Bears on

• Problem 507: only through its case $s=0$, Theorem 1.1, which leads to the paper's Theorem 1.8.