论文标题
线性cantor套件的Minkowski总和
The Minkowski sum of linear Cantor sets
论文作者
论文摘要
令$ c $为经典的中部第三大康托套件。众所周知，$ C+C = [0,2] $（Steinhaus，1917）。 （在这里$ + $表示Minkowski sum。）令$ u $为[0,2] $的$ z \ set，其唯一表示为$ z = x + y $，带有$ x，y \ in c $（一组独特）。并不难证明$ \ dim_h u = \ log（2） / \ log（3）$和$ u $本质上看起来像$ 2C $。 假设$ 0，n-1 \在A \ subset \ {0,1，\ dots，n-1 \} $中，将$ c_a = c_a = c_a = c_a {a，n} $作为迭代函数系统吸引子\ [\ [\ {x \ mapsto（x + a） / n：a \}的线性cantor集。 \]我们考虑此类线性cantor集的各种属性。我们的主要重点是$ c_ {a，n}+c_ {a，n} $的结构，具体取决于$ n $和$ a $，以及一组唯一性$ u_a $的属性。
Let $C$ be the classical middle third Cantor set. It is well known that $C+C = [0,2]$ (Steinhaus, 1917). (Here $+$ denotes the Minkowski sum.) Let $U$ be the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y \in C$ (the set of uniqueness). It isn't difficult to show that $\dim_H U = \log(2) / \log(3)$ and $U$ essentially looks like $2C$. Assuming $0,n-1 \in A \subset \{0,1,\dots,n-1\}$, define $C_A = C_{A,n}$ as the linear Cantor set which the attractor of the iterated function system \[ \{ x \mapsto (x + a) / n: a \in A \}. \] We consider various properties of such linear Cantor sets. Our main focus will be on the structure of $C_{A,n}+C_{A,n}$ depending on $n$ and $A$ as well as the properties of the set of uniqueness $U_A$.
