论文标题
紧凑型套件，带有大型预测，无处浓密的集合
Compact sets with large projections and nowhere dense sumset
论文作者
论文摘要
我们回答了Banakh，Jabłońska和Jabłoński的一个问题，该问题表明，对于$ d \ ge 2 $，存在一个紧凑的套装$ k \ subseteq \ subseteq \ mathbb {r}^d $，因此$ k $的投影$ k $的投影在每个增生的内部均为非空地上，但$ K+K+K+现在是$ K+K $。证明依赖于随机结构。 证明的一种自然方法是在具有完整预测的单位立方体中构建这样的$ k $，即$ k $的预测与单位立方体的预测一致。我们研究了这些问题对各个维子空间的投影的概括以及$ \ ell $折叠的总和。我们获得了许多积极和负面的结果，但也留下了许多有趣的案例。 我们还表明，在大多数情况下，如果我们有这样的紧凑型集的特定示例，那么实际上（从Baire类别的意义上）则在适当选择的空间中设置了一个固定套件，也就是一个示例。 最后，利用计算机辅助结构，我们证明了飞机上的紧凑型设置，带有完整的投影，无处可相似。
We answer a question of Banakh, Jabłońska and Jabłoński by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a $K$ in the unit cube with full projections, that is, such that the projections of $K$ agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for $\ell$-fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.
