用管子覆盖谢尔皮耶斯基地毯
Covering the Sierpiński carpet with tubes
摘要：
我们表明，非平凡的$\times N$不变集在$[0,1]^d$，如Sierpienski地毯和Sierpienski海绵，管空，也就是说，他们可以覆盖的一个工会的管状邻域线的任意小的总体积。本文引入了一类新的维数严格介于$d-1$和$d$之间的管-空集。我们利用遍历理论的方法将集合分解为多个部分，每个部分在某个方向上投影到一个豪斯多夫维数小于$1$的集合上。我们还讨论了其他自相似集的管覆盖，并提出了各种应用。
英文摘要：
We show that non-trivial $\times N$-invariant sets in $[0,1]^d$, such as the Sierpiński carpet and the Sierpiński sponge, are tube-null, that is, they can be covered by a union of tubular neighbourhoods of lines of arbitrarily small total volume. This introduces a new class of tube-null sets of dimension strictly between $d-1$ and $d$. We utilize ergodic-theoretic methods to decompose the set into finitely many parts, each of which projects onto a set of Hausdorff dimension less than $1$ in some direction. We also discuss coverings by tubes for other self-similar sets, and present various applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
