Kakeya conjectures under the Polynomial Wolff Axioms

Establish the Kakeya bounds for families of δ-tubes in R^n that satisfy the Polynomial Wolff Axioms: (A) show |⋃_T T| ≳≈ ∑_T |T|; (B) for shadings Y(T) ⊂ T with |Y(T)| ≥ (log(1/δ))^{-1}|T|, show |⋃_T Y(T)| ≳≈ ∑_T |T|; and (C) show ||∑_T χ_T||_{n/(n−1)} ≲≈ 1.

Background

In dimensions n ≥ 4, varieties such as quadric hypersurfaces can contain many lines and produce counterexamples to Kakeya-type lower bounds under only convex non-concentration. The Polynomial Wolff Axioms strengthen non-clustering to exclude concentration in semi-algebraic sets.

These conjectures aim to recover Kakeya lower bounds and maximal function control for tube families satisfying the more robust algebraic non-concentration, extending implications for Minkowski/Hausdorff dimension and maximal operator bounds.

References

Conjecture\label{KakeyaForPolynomialWolff} (A): Let $T$ be a set of $\delta$ tubes in $Rn$ that satisfy the Polynomial Wolff Axioms. Then

\Big|\bigcup_{T}T\Big| \gtrapprox \sum_T |T|.

(B): For each $T\inT$, let $Y(T)\subset T$ with $|Y(T)|\geq (\log 1/\delta){-1} |T|$. Then

\Big|\bigcup_{T}T\Big| \gtrapprox \sum_T |T|.

(C):

\Big\Vert \sum_{T\inT}\chi_T\Big\Vert_{\frac{n}{n-1}\lessapprox 1.

— A Survey of the Kakeya conjecture, 2000-2025 (2512.09397 - Zahl, 10 Dec 2025) in Conjecture \ref{KakeyaForPolynomialWolff}, Section 4.1