论文标题
典型康托套件的所有预测都是康托尔集
All projections of a typical Cantor set are Cantor sets
论文作者
论文摘要
在1994年，约翰·科布（John Cobb）问：给定$ n> m> k> 0 $，是否存在$ \ mathbb r^n $中的cantor设置，以使其每个投影均为$ m $ planes恰好是$ k $ dipermensional？ L.Antoine（1924）和K.Borsuk（1947）以$（N，M，K）的形式描述了（N，M，K）=（2,1,1）$。 J.Cobb（1994）为$（3,2,1）$构建了示例，以$（N，M，M，M-1）$的价格构建了示例，以及O.Frolkina（2010，2019）的$（n，n-n-1，n-2）$以不同的方式，以$（N，N-1，K）的价格为$（N，N-1，N-2）$。我们表明，从以下意义上讲，这种集合是例外的。令$ \ Mathcal C（\ Mathbb r^n）$为$ \ Mathbb r^n $的所有cantor子集，该子集已符合Hausdorff Metric。众所周知，$ \ MATHCAL C（\ MATHBB R^n）$是Baire空间。我们证明，有一个密集的$g_δ$ subset $ \ mathcal p \ subset \ mathcal c（\ mathbb r^n）$，使得每个$ x \ in \ in \ mathcal p $ in \ mathcal p $和每个非零线性subspace $ l \ l \ subset $ l \ subseT \ subbb r^n $，$ x $ a $ x $ a $ x $ a $ a a cant $ x $ a a cant l o a $ x $ a aptor。这给出了同一论文（1994）中J.COBB的另一个问题的部分答案。
In 1994, John Cobb asked: given $N>m>k>0$, does there exist a Cantor set in $\mathbb R^N$ such that each of its projections into $m$-planes is exactly $k$-dimensional? Such sets were described for $(N,m,k)=(2,1,1)$ by L.Antoine (1924) and for $(N,m,m)$ by K.Borsuk (1947). Examples were constructed for the cases $(3,2,1)$ by J.Cobb (1994), for $(N,m,m-1)$ and in a different way for $(N,N-1,N-2)$ by O.Frolkina (2010, 2019), for $(N,N-1,k)$ by S.Barov, J.J.Dijkstra and M.van der Meer (2012). We show that such sets are exceptional in the following sense. Let $\mathcal C(\mathbb R^N)$ be a set of all Cantor subsets of $\mathbb R^N$ endowed with the Hausdorff metric. It is known that $\mathcal C(\mathbb R^N)$ is a Baire space. We prove that there is a dense $G_δ$ subset $\mathcal P \subset \mathcal C(\mathbb R^N)$ such that for each $X\in \mathcal P$ and each non-zero linear subspace $L \subset \mathbb R^N$, the orthogonal projection of $X$ into $L$ is a Cantor set. This gives a partial answer to another question of J.Cobb stated in the same paper (1994).
