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arXiv 1701.07045math.CA
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多项式Wolff公理与$\mathbb{R}^4$中的Kakeya型估计

Polynomial Wolff axioms and Kakeya-type estimates in $\mathbb{R}^4$

Larry Guth, Joshua Zahl

备注：
30 pages, 0 figures. v3: An error has been corrected, which weakens the dimension of our maximal function estimate to 3 + 1/40. Previously, a bound of 3 + 1/28 was claimed. See the addendum following the statement of Theorem 1.1 for details
发表信息：
Proc. London Math. Soc. 117(1): 192--220, 2018
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摘要：

我们建立了新的线性和三线性的界限集合管$\mathbb{R}^4$，满足多项式沃尔夫公理。简而言之，如果平面的$δ$ -邻域中不能包含太多的管，则$δ$ -管的集合满足沃尔夫公理。如果在一个低次代数簇的$δ$ -邻域中不能包含太多的管，则管的集合满足多项式Wolff公理。首先，我们证明了如果$\mathbb{R}^4$中的一组$δ^{-3}$管满足多项式Wolff公理，则这些管的并必须具有至少$δ^{1-1/40}$的体积。我们还证明了一个更技术性的声明，这是类似于一个最大的功能估计在尺寸$3+1/40$。其次，证明了如果$\mathbb{R}^4$中的$δ^{-3}$管集合满足多项式Wolff公理，并且如果相交管的大多数三元组指向三个线性无关的方向，则管的并集的体积至少为$δ^{3/4}$ .同样，我们还证明了一个稍微技术性的声明，这是类似于一个最大的功能估计尺寸$3+1/4$。我们猜想，每个Kakeya集满足多项式沃尔夫公理，但我们无法证明这一点。如果我们的猜想是正确的，它意味着一个Kakeya极大函数估计在维度$3+1/40$，特别是这意味着每个Kakeya集在$\mathbb{R}^4$必须有豪斯多夫维数至少$3+1/40$。这将是对沃尔夫在1995年建立的当前最佳界限3的改进。

英文摘要：

We establish new linear and trilinear bounds for collections of tubes in $\mathbb{R}^4$ that satisfy the polynomial Wolff axioms. In brief, a collection of $δ$-tubes satisfies the Wolff axioms if not too many tubes can be contained in the $δ$-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the $δ$-neighborhood of a low degree algebraic variety. First, we prove that if a set of $δ^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least $δ^{1-1/40}$. We also prove a more technical statement which is analogous to a maximal function estimate at dimension $3+1/40$. Second, we prove that if a collection of $δ^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least $δ^{3/4}$. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension $3+1/4$. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension $3+1/40$, and in particular this implies that every Kakeya set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/40$. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.

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