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Source. Theorem 1.9, p. 6, with the definitions on pp. 5-7 and their restatement in §5.1 (p. 30), 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 pages. The proof (§5.1, from Theorem 5.1 and Proposition 4.4, with Theorem 5.1 proved in §6) was read for structure only and is not checked here.

Statement

Setting (pp. 5-7). A point $p=(a,b)$ with a line $\ell$ through it of slope $c\in[-1,1]$ is the phase-space point $(a,b,c)\in\Omega=[-1,1]^3$; for $\omega=(a,b,c)$ write $p_\omega=(a,b)$ and $\ell_\omega=\{(a+t,b+ct):t\in\mathbb R\}$. The phase-space rectangle of dimensions $u\times uw\times w$ centered at $(a_0,b_0,c_0)$ is $\{(a_0+t,\,b_0+c_0t+r,\,c_0+s):(t,r,s)\in[-u,u]\times[-uw,uw]\times[-w,w]\}$ (equation (6), p. 5). A set $\mathbf X\subset\Omega$ is a $(\delta,\alpha,\beta,C)$-set if it is $\delta$-separated in $\mathbb R^3$ and $|\mathbf X\cap\mathbf R|\le Cu^\alpha w^\beta|\mathbf X|$ for every $u\times uw\times w$ rectangle $\mathbf R$ with $uw\ge\delta$ (p. 5); the restatement on p. 30 takes $u,w\in(0,1]$. The smoothed incidence count is $I(\delta;\mathbf X)=\sum_{\omega_1,\omega_2\in\mathbf X}\eta(d(p_{\omega_1},\ell_{\omega_2})/\delta)$ (p. 6), the sum over all ordered pairs, where $\eta:\mathbb R\to[0,1]$ is a fixed smooth bump with support in $[-1/2-1/10,1/2+1/10]$, equal to $1$ on $[-1/2+1/10,1/2-1/10]$ and with integral $1$, chosen so that the high-low inequality (Theorem 1.7, p. 3) holds (pp. 7-8).

Theorem 1.9 (p. 6, quoted). "Let $\alpha,\beta\in[1,2]$ satisfy $\alpha+\beta>3$, and let $\varepsilon>0$. There exists $\eta=\eta(\alpha,\beta,\varepsilon)>0$ such that the following holds for all $\delta<\delta_0(\alpha,\beta,\varepsilon)$. Suppose $\mathbf{X}\subset\Omega$ is a $(\delta,\alpha,\beta,\delta^{-\eta})$-set. Then $I(\delta;\mathbf{X})\geqslant\delta^{1+\varepsilon}|\mathbf{X}|^2$."

The paper calls this its main result (pp. 7, 8). It remarks (p. 6) that $\delta|\mathbf X|^2$ is the expected count for randomly placed points and lines, that the matching upper bound $I(\delta;\mathbf X)\le\delta^{1-\varepsilon}|\mathbf X|^2$ follows from the high-low inequality under the same hypotheses, and that for every $\gamma<3/2$ some $(\delta,\gamma,\gamma,C)$-set has $I(\delta;\mathbf X)\ge\delta^{1-\varepsilon}|\mathbf X|^2$ with $\varepsilon=\varepsilon(\gamma)>0$, built from the Szemerédi-Trotter example (footnote 3, p. 6), so the upper bound fails there.

Proof pointer

§5.1 (p. 30), by contradiction and compactness: a failing sequence of sets has a limiting branching function in the class $\mathcal L_{\alpha,\beta}$, which Theorem 5.1 (proved in §6) places in the class $\mathcal L^{good}$, and Proposition 4.4 then gives the incidence bound. §2 (from p. 8) sketches the two-step method: an initial estimate at a coarse scale and an inductive step from the high-low inequality.

Dependencies

Theorem 1.7, Proposition 4.4 and Theorem 5.1 of the same paper.

Bears on

• Problem 507: only through Theorem 1.4, Theorem 1.1 and Corollary 1.2, which lead to the paper's Theorem 1.8.