Wang–Wu's two-ends Furstenberg conjecture

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Let δ∈(0,1)\delta\in(0,1). Let (L,Y)δ(L,Y)_\delta be a set of directional δ\delta-separated lines in Rn\mathbb{R}^n with an (ε1,ε2,CY)(\varepsilon_1,\varepsilon_2,C_Y)-two-ends, λ\lambda-dense shading at scale δ\delta. Wang–Wu's two-ends Furstenberg conjecture. For any ε>0\varepsilon>0, there is cεc_\varepsilon, also depending on ε1\varepsilon_1, ε2\varepsilon_2 and CYC_Y, such that

∣⋃ℓ∈LY(ℓ)∣≥cεδεδO(ε1)λn−12∑ℓ∈L∣Y(ℓ)∣.\left|\bigcup_{\ell\in L}Y(\ell)\right|\geq c_\varepsilon\delta^{\varepsilon}\delta^{O(\varepsilon_1)}\lambda^{\frac{n-1}{2}}\sum_{\ell\in L}|Y(\ell)|.

This conjecture is attributed to Wang and Wu and concerns lower bounds for unions of shaded, directionally separated lines; the supplied text gives no evidence that it has been resolved.

References

Primary source

Ciprian Demeter and Shukun Wu, “Restriction and decoupling estimates for the hyperbolic paraboloid in R^3”, arXiv:2505.09037 (2026).

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