Assume $0<r_0<n$. Are there sets $A,B\subseteq \mathbb{R}^n$, such that the Hasudorff dimension of $A,B$ are zero, But $\dim_H(A+B)=r_0$?
When $r_0$ is integer, I have found(By attention to page 104,the example 7.8 of the book FRACTAL GEOMETRY Mathematical Foundations and Applications, version 2,Kenneth Falconer ) the $A,B$ with the above property. But when $r_0$ is not integer, I don't know how to find these sets?