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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2411.14861 |
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- Suppose that $K$ and $ K'$ are two affine Cantor sets. It is shown that the sum set $K+K'$ has equal box and Hausdorff dimensions and in this number named $s$, $H^s(K+K')<\infty$. Moreover, for almost every pair $(K,K')$ satisfying $HD(K)+HD(K')\leq 1$, there is a dense subset $D\subset \mathbb R$ such that $H^s(K+λK')=0$, for all $λ\in D$. It also is shown that in the context of affine Cantor sets with two increasing maps, there are generically (topological and almost everywhere) five possible structures for their sum: a Cantor set, an L, R, M-Cantorval or a finite union of closed intervals.