论文标题
具有大量大型预测的集合的分析能力和尺寸
Analytic capacity and dimension of sets with plenty of big projections
论文作者
论文摘要
我们的主要结果标志着旧的Vitushkin猜想的进展。我们表明，在平面上有大量大型投影（PBP）具有正分析能力，以及定量下限。还证明了与Riesz内核有关的能力，包括Lipschitz谐波容量。该证明使用较低内容的常规集合使用弗罗斯特曼（Frostman）的构造构造，这可能引起独立的兴趣。我们的第二个主要结果是分析师的旅行推销员定理，旨在进行大量预测。作为推论，我们获得了用PBP统一摇摆的Hausdorff尺寸的下限。第二个推论是针对Denjoy的猜想的定量解决方案的精神，对使用PBP的子集的容量估计。
Our main result marks progress on an old conjecture of Vitushkin. We show that a compact set in the plane with plenty of big projections (PBP) has positive analytic capacity, along with a quantitative lower bound. A higher dimensional counterpart is also proved for capacities related to the Riesz kernel, including the Lipschitz harmonic capacity. The proof uses a construction of a doubling Frostman measure on a lower content regular set, which may be of independent interest. Our second main result is the Analyst's Traveling Salesman Theorem for sets with plenty of big projections. As a corollary, we obtain a lower bound for the Hausdorff dimension of uniformly wiggly sets with PBP. The second corollary is an estimate for the capacities of subsets of sets with PBP, in the spirit of the quantitative solution to Denjoy's conjecture.
