论文标题
用管子覆盖Sierpiński地毯
Covering the Sierpiński carpet with tubes
论文作者
论文摘要
我们表明，$ [0,1]^d $的非平凡$ \ times n $ invariant套件，例如Sierpiński地毯和SierpińskiSponge，即管null，也就是说，它们可以被任意小体积的小管群落所覆盖。这引入了一类新的管子无尺寸，严格在$ d-1 $和$ d $之间。我们利用千古理论方法将集合分解为有限的许多部分，每个部分都将其投影到一组Hausdorff尺寸上，而在某个方向上小于$ 1 $。我们还讨论了针对其他自相似集合的覆盖物，并介绍了各种应用。
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.
