论文标题
在$ {\ bbb r}^d $中的分形集的算术总和
On arithmetic sums of fractal sets in ${\Bbb R}^d$
论文作者
论文摘要
如果存在一个正整数$ n $，则紧凑型集合$ e \ subset {\ bbb r}^d $在算术上是厚度厚的，因此$ n $ fold-fold $ fold $ e $ $ e $具有非空的内饰。 We prove the arithmetic thickness of $E$, if $E$ is uniformly non-flat, in the sense that there exists $ε_0>0$ such that for $x\in E$ and $0<r\leq {\rm diam}(E)$, $E\cap B(x,r)$ never stays $ε_0r$-close to a hyperplane in ${\Bbb r}^d $。此外，我们证明了几类分形组的算术厚度，包括自相似集合，$ {\ bbb r}^d $（带有$ d \ geq 2 $）的自符号集和$ {\ bb bb raf}^2 $ bb bb的$ $ fraff $ $ fraff $ nmbb n in hexplane sepflane n in hexplane sepfrane和bb的自我交付套件（ （带有$ d \ geq 3 $）在特定假设下。
A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly non-flat, in the sense that there exists $ε_0>0$ such that for $x\in E$ and $0<r\leq {\rm diam}(E)$, $E\cap B(x,r)$ never stays $ε_0r$-close to a hyperplane in ${\Bbb R}^d$. Moreover, we prove the arithmetic thickness for several classes of fractal sets, including self-similar sets, self-conformal sets in ${\Bbb R}^d$ (with $d\geq 2$) and self-affine sets in ${\Bbb R}^2$ that do not lie in a hyperplane, and certain self-affine sets in ${\Bbb R}^d$ (with $d\geq 3$) under specific assumptions.
