All Questions

In a recent paper, we required the following fact. Proposition 1. There exists a simple closed curve $\gamma\subset\mathbb{C}$ with the following property. If $\phi$ is a biholomorphic map, defined on ...

Can there be a set whose Hausdorff dimension gradually changes? For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ...

Previously asked at MSE: Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to tweak that if it would lead to a more ...

Edit: According to comment of Andre Henriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...

I am thinking of the following question: Let $X\subseteq \mathbb R$. Is it true that $$ \mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)? $$ My thoughts: We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(...