论文标题
与算术Kakeya猜想有关的模式问题
Pattern Problems related to the Arithmetic Kakeya Conjecture
论文作者
论文摘要
我们研究了有关与Kakeya猜想有关的组合的各种问题。特别是，我们表明其中许多问题等同于Kakeya的Kakeya猜想。我们还提供了一个证据，表明算术Kakeya的猜想意味着Kakeya的猜想是包装维度的，因为这种含义以前仅以Minkowski维度而闻名。 我们考虑了几个问题，类似于Stein和Bourgain的经典结果，内容涉及包含$ [0,1]^n $的球体的集合的Lebesgue度量，我们将通过任意多层型代替球体。我们在$ \ mathbb {r}^n $中给出了多层的下限，并证明这对于简单而言是锋利的。最后，我们将绿色和Ruzsa的数字理论方法概括为研究数字领域的模式，从而为这些同类问题中的几个问题提供了上限。
We study a variety of problems about homothets of sets related to the Kakeya conjecture. In particular, we show many of these problems are equivalent to the arithmetic Kakeya conjecture of Katz and Tao. We also provide a proof that the arithmetic Kakeya conjecture implies the Kakeya conjecture for packing dimension, as this implication was previously only known for Minkowski dimension. We consider several questions analogous to the classical results of Stein and Bourgain about the Lebesgue measure of a set containing a sphere centered at every point of $[0,1]^n$, where we replace spheres by arbitrary polytopes. We give a lower bound for polytopes in $\mathbb{R}^n$, and show that this is sharp for simplices. Finally, we generalize number theoretic methods of Green and Ruzsa to study patterns in number fields and thereby provide upper bounds for several of these homothet problems.
