A compact set $E\subset {\mathbb{R}}^d$ 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 ${\epsilon }_0>0$ such that for $x\in E$ and $0<r⩽\mathrm{diam}(E)$, $E\cap B(x,r)$ never stays ${\epsilon }_{0}r$-close to a hyperplane in ${\mathbb{R}}^d$. Moreover, we prove the arithmetic thickness for several classes of fractal sets, including self-similar sets, self-conformal sets in ${\mathbb{R}}^d$ (with $d⩾2$) and self-affine sets in ${\mathbb{R}}^2$ that do not lie in a hyperplane, and certain self-affine sets in ${\mathbb{R}}^d$ (with $d⩾3$) under specific assumptions.

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关于 Rd 中分形集的算术和

紧凑的套装 $乙\subset {\mathbb{电阻}}^d$ 如果存在正整数，则称其为算术厚 $n$ 所以这样 $n$-fold 的算术和 $乙$有非空的内部。我们证明了算术厚度$乙$， 如果 $乙$ 是一致非平坦的，因为存在 ${\epsilon }_0>0$ 使得对于 $X\in 乙$ 和 $0<r⩽\mathrm{直径}(乙)$, $乙\cap 乙(X,r)$ 从不停留 ${\epsilon }_{0}r$- 接近超平面 ${\mathbb{电阻}}^d$. 此外，我们证明了几类分形集的算术厚度，包括自相似集，自共形集${\mathbb{电阻}}^d$ （和 $d⩾2$) 和自仿射集 ${\mathbb{电阻}}^2$ 不位于超平面中，并且某些自仿射设置在 ${\mathbb{电阻}}^d$ （和 $d⩾3$) 在特定假设下。