Discretized Kakeya conjecture (union measure lower bounds)

Establish the discretized Kakeya lower bounds in R^n: (A) show that for any set T of δ-tubes pointing in δ-separated directions, the measure of the union satisfies |⋃_T T| ≳≈ ∑_T |T|; and (B) show that for measurable shadings Y(T) ⊂ T with |Y(T)| ≥ (log(1/δ))^{-1} |T|, the union satisfies |⋃_T Y(T)| ≳≈ ∑_T |T|.

Background

This discretized formulation quantifies the small-scale behavior of Besicovitch sets, connecting Minkowski and Hausdorff dimension to volume lower bounds for unions of thin tubes oriented in separated directions.

Part (A) implies the Minkowski dimension version of Kakeya, while Part (B) implies the Hausdorff dimension version. These bounds are central to modern approaches via induction on scale and geometric combinatorics.

References

Conjecture \ref{KakeyaDiscretized} (A): Let $T$ be a set of $\delta$ tubes in $Rn$ pointing in $\delta$-separated directions. Then

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

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

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

— A Survey of the Kakeya conjecture, 2000-2025 (2512.09397 - Zahl, 10 Dec 2025) in Conjecture \ref{KakeyaDiscretized}, Section 1 (Introduction)